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Participants in 

Mathematics Teacher Education 

Individuals, Teams, 
Communities and Networks 

Konrad Krainer and Terry Wood (Eds.) 

The International Handbook of Mathematics 
Teacher Education 

Series Editor: 

Terry Wood 
Purdue University 
West Lafayette 

This Handbook of Mathematics Teacher Education, the first of its kind, addresses the learning of 
mathematics teachers at all levels of schooling to teach mathematics, and the provision of activity and 
programmes in which this learning can take place. It consists of four volumes. 


Knowledge and Beliefs in Mathematics Teaching and Teaching Development 

Peter Sullivan, Monash University, Clayton, Australia and Terry Wood, Purdue University, West 

Lafayette, USA (eds.) 

This volume addresses the "what" of mathematics teacher education, meaning knowledge for 

mathematics teaching and teaching development and consideration of associated beliefs. As well as 

synthesizing research and practice over various dimensions of these issues, it offers advice on best 

practice for teacher educators, university decision makers, and those involved in systemic policy 

development on teacher education. 

paperback: 978-90-8790-541-5, hardback: 978-90-8790-542-2, ebook: 978-90-8790-543-9 


Tools and Processes in Mathematics Teacher Education 

Dina Tirosh, Tel Aviv University, Israel and Terry Wood, Purdue University, West Lafayette, USA 


This volume focuses on the "how" of mathematics teacher education. Authors share with the readers 

their invaluable experience in employing different tools in mathematics teacher education. This 

accumulated experience will assist teacher educators, researchers in mathematics education and those 

involved in policy decisions on teacher education in making decisions about both the tools and the 

processes to be used for various purposes in mathematics teacher education. 

paperback: 978-90-8790-544-6, hardback: 978-90-8790-545-3, ebook: 978-90-8790-546-0 


Participants in Mathematics Teacher Education: Individuals, Teams, Communities and Networks 

Konrad Krainer, University of Klagenfurt, Austria and Terry Wood, Purdue University, West Lafayette, 

USA (eds.) 

This volume addresses the "who" question of mathematics teacher education. The authors focus on the 

various kinds of participants in mathematics teacher education, professional development and reform 

initiatives. The chapters deal with prospective and practising teachers as well as with teacher educators 

as learners, and with schools, districts and nations as learning systems. 

paperback: 978-90-8790-547-7, hardback: 978-90-8790-548-4, ebook: 978-90-8790-549-1 


The Mathematics Teacher Educator as a Developing Professional 

Barbara Jaworski, Loughborough University, UK and Terry Wood, Purdue University, West Lafayette, 

USA (eds.) 

This volume focuses on knowledge and roles of teacher educators working with teachers in teacher 

education processes and practices. In this respect it is unique. Chapter authors represent a community 

of teacher educators world wide who can speak from practical, professional and theoretical viewpoints 

about what it means to promote teacher education practice. 

paperback: 978-90-8790-550-7, hardback: 978-90-8790-551-4, ebook: 978-90-8790-552-1 

Participants in Mathematics 
Teacher Education 

Individuals, Teams, Communities and Networks 

Edited by 

Konrad Krainer 

University of Klagenfurt, Austria 


Terry Wood 

Purdue University, West Lafayette, USA 


^jjfe CICATA - IPN 
* * 



A C.I. P. record for this book is available from the Library of Congress. 

ISBN 978-90-8790-547-7 (paperback) 
ISBN 978-90-8790-548-4 (hardback) 
ISBN 978-90-8790-549-1 (e-book) 

Published by: Sense Publishers, 

P.O. Box 21858, 3001 AW Rotterdam, The Netherlands 

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Cover picture: t ^^..„^^^ s „ ._...- 

Badell river, Tavernes de la Valldigna, Valencia, Spain 
© Pepa and Ana Llinares 2007 

One drop of water does not make a river, yet each drop bears in itself the full fluidity and power of 
water. A river is more than millions of drops, it is a large and vital system. It represents an astonishing 
journey from the source to its mouth at the sea, from the micro to the macro level. A river marks a 
steady coming together and growing but also finding different branches and maybe courses. It depends 
on and is influenced by external elements like rain and rocks, but also by pollution. It is formed by its 
environment, but is in turn a force upon its environment. A river is a journey of necessary collaboration 
in a joint process. 
© Salvador Llinares and Konrad Krainer 2008 

Printed on acid-free paper 

All rights reserved © 2008 Sense Publishers 

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by 
any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written 
permission from the Publisher, with the exception of any material supplied specifically for the purpose 
of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 


Preface ix 

Individuals, Teams, Communities and Networks: Participants and Ways 

of Participation in Mathematics Teacher Education: An Introduction 1 

Konrad Krainer 

Section 1: Individual Mathematics Teachers as Learners 

Chapter 1 : Individual Prospective Mathematics Teachers: Studies on 

Their Professional Growth 13 

Helia Oliveira and Markku S. Hannula 

Chapter 2: Individual Practising Mathematics Teachers: Studies on 

Their Professional Growth 35 

Marie-Jeanne Perrin-Glorian, Lucie DeBlois, and Aline Robert 

Section 2: Teams of Mathematics Teachers as Learners 

Chapter 3: Teams of Prospective Mathematics Teachers: Multiple Problems 

and Multiple Solutions 63 

Roza Leikin 

Chapter 4: Teams of Practising Teachers: Developing Teacher Professionals 89 
Susan D. Nickerson 

Chapter 5: Face-to-Face Learning Communities of Prospective Mathematics 
Teachers: Studies on Their Professional Growth 1 1 1 

Fou-Lai Lin andJoao Pedro da Ponte 

Section 3: Communities and Networks of Mathematics Teachers as Learners 

Chapter 6: Face-to-Face Communities and Networks of Practising 

Mathematics Teachers: Studies on Their Professional Growth 133 

Stephen Lerman and Stefan Zehetmeier 


Chapter 7: Virtual Communities and Networks of Prospective Mathematics 
Teachers: Technologies, Interactions and New Forms of Discourse 1 55 

Salvador Llinares and Federica Olivero 

Chapter 8: Virtual Communities and Networks of Practising Mathematics 
Teachers: The Role of Technology in Collaboration 1 8 1 

Marcelo Borba and George Gadanidis 

Section 4: Schools, Regions and Nations as Mathematics Learners 

Chapter 9: School Development as a Means of Improving Mathematics 
Teaching and Learning: Towards Multidirectional Analyses of Learning 
across Contexts 209 

Elham Kazemi 

Chapter 10: District Development as a Means of Improving Mathematics 
Teaching and Learning at Scale 23 1 

Paul Cobb and Thomas Smith 

Chapter 1 1 : Studies on Regional and National Reform Initiatives as a Means 

to Improve Mathematics Teaching and Learning at Scale 255 

John Peggand Konrad Krainer 

Section 5: Teachers and Teacher Educators as Key Players in the 
Further Development of the Mathematics Teaching Profession 

Chapter 12: The Use of Action Research in Teacher Education 283 

Gertraud Benke, Alena Hospesovd, and Marie Tichd 

Chapter 13: Building and Sustaining Inquiry Communities in Mathematics 
Teaching Development: Teachers and Didacticians in Collaboration 309 

Barbara Jaworski 

Chapter 14: Mathematics Teaching Profession 33 1 

Nanette Seago 

Section 6: Critical Respondants 

Chapter 15: Pathways in Mathematics Teacher Education: Individual 

Teachers and Beyond 355 

Gilah C. Leder 



Chapter 16: Individuals, Teams and Networks: Fundamental Constraints of 
Professional Communication Processes of Teachers and Scientists about 
Teaching and Learning Mathematics 369 

Heinz Steinbring 



It is my honor to introduce the first International Handbook of Mathematics 
Teacher Education to the mathematics education community and to the field of 
teacher education in general. For those of us who over the years have worked to 
establish mathematics teacher education as an important and legitimate area of 
research and scholarship, the publication of this handbook provides a sense of 
success and a source of pride. Historically, this process began in 1987 when 
Barbara Jaworski initiated and maintained the first Working Group on mathematics 
teacher education at PME. After the Working Group meeting in 1994, Barbara, 
Sandy Dawson and I initiated the book, Mathematics Teacher Education: Critical 
International Perspectives, which was a compilation of the work accomplished by 
this Working Group. Following this, Peter de Liefde who, while at Kluwer 
Academic Publishers, proposed and advocated for the Journal of Mathematics 
Teacher Education and in 1998 the first issue of the journal was printed with 
Thomas Cooney as editor of the journal who set the tone for quality of manuscripts 
published. From these events, mathematics teacher education flourished and 
evolved as an important area for investigation as evidenced by the extension of 
JMTE from four to six issues per year in 2005 and the recent 15 th ICMI Study, The 
professional education and development of teachers of mathematics. In preparing 
this handbook it was a great pleasure to work with the four volume editors, Peter 
Sullivan, Dina Tirosh, Konrad Krainer and Barbara Jaworski and all of the authors 
of the various chapters found throughout the handbook. 

Volume 3, Participants in mathematics teacher education: Individuals, teams, 
communities and networks, edited by Konrad Krainer, focuses not only on 
individual prospective and practicing teachers as learners but also on teams, 
learning communities, networks of teachers, schools and on the teaching profession 
as a whole. In this volume, the emphasis is on describing and critically analysing 
participants' organizational contexts. This is the third volume of the handbook. 

Terry Wood 
West Lafayette, IN 


Jaworski, B., Wood, T., & Dawson, S. (Eds). (1999). Mathematics teacher education: Critical 

international perspectives. London: Falmer Press. 
Krainer, K., & Wood, T. (Eds.). (2008). International handbook of mathematics teacher education: Vol. 

3. Participants in mathematics teacher education: Individuals, teams, communities and networks. 

Rotterdam, the Netherlands: Sense Publishers. 
Wood, T. (Series Ed), Jaworski, B., Krainer, K., Sullivan, P., & Tirosh, D. (Vol. Eds.). (2008). 

International handbook of mathematics teacher education. Rotterdam, the Netherlands: Sense 








An Introduction 

This chapter outlines the main idea of the third volume of this handbook. The focus 
is to make transparent the large diversity of experiences evident in the field of 
mathematics teacher education and to describe and learn from these differences. 
This is achieved by focusing on the practices of various teacher education 
participants and the environments in which they work. Notions like individuals, 
teams, communities and networks are introduced. In addition, school development 
as well as regional and national teacher education initiatives are regarded. Each 
chapter considers differences within and among these distinctions. 


What is (mathematics) teacher education and who are its participants! The teacher 
educator can be regarded as the teacher of prospective or practising teachers. 
Although, traditionally, the latter are regarded mainly as the participants and 
learners, also the teacher educator is a part of the learning process and grows 
professionally from participating in the process. Therefore, it makes sense to regard 
student teachers, teachers and teacher educators as teachers and as (lifelong) 
learners at the same time (see also Llinares & Krainer, 2006). Teachers are active 
constructors of their knowledge, embedded in a variety of social environments. 
These environments influence and shape teachers' beliefs, knowledge and practice; 
similarly, teachers themselves influence and shape their environments. Therefore, 
teachers should be expected to continuously reflect in and on their practice and the 
environments in which they work; teachers should also be prepared to make 
changes where it is appropriate. Whereas teachers and students in a classroom (at a 
school or in a course for prospective teachers) mostly show considerable 
differences in age and experience, the relation in the context of professional 
development activities might be more diverse (e.g., having a young researcher 
working with experienced teachers). 

Teacher education is a goal-directed intervention in order to promote teachers' 
learning, including all formal kinds of teacher preparation and professional 
development as well as informal (self-organized) activities. Mathematics teacher 

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 1-10. 
© 2008 Sense Publishers. All rights reserved. 


education can aim at improving teachers' beliefs, their knowledge and their 
practice, at increasing their motivation, their self-confidence and their identity as 
mathematics teachers and, most importantly, at contributing to their students' 
affective and cognitive growth. 

Although the education of prospective mathematics teachers is organized in a 
wide variety of ways in different countries and at different teacher education 
institutes, teacher education activities for practicing mathematics teachers are even 
more diverse. They include, for example: formal activities led by externals (e.g., by 
mathematics educators and mathematicians) or informal and self-organized ones 
(e.g., by a group of mathematics teachers themselves); single events or continuous 
and long-lasting programmes; small-group courses or nation-wide mathematics 
initiatives (eventually with hundreds of participants); heterogeneous groups of 
participants (e.g., mathematics and science teachers from all parts of a country) or 
a mathematics-focused school development programme at one single school; 
obligatory participation in courses or voluntary engagement in teacher networks; 
focus on specific contents (e.g., early algebra) or on more general issues (e.g., new 
modes of instruction); theory-driven seminars at universities or teaching 
experiments at schools; focus on primary or secondary schooling; teacher 
education accompanied by extensive research or confined to minimal evaluation; 
activities that aim at promoting teachers' (different kinds of) knowledge, or beliefs, 
or practice etc. 

It is a challenge to find answers to the questions of where, under which 
conditions, how and why mathematics teachers learn and how important the 
domain-specific character of mathematics is. It is important to take into account 
that teachers' learning is a complex process and is to a large extent influenced by 
personal, social, organisational, cultural and political factors. 

Discussions about "effective", "good", "successful" etc. mathematics teaching 
or teacher education indicate different aspects which need to be considered (see 
e.g., Sowder, 2007). However, in any case the dimensions contents, communities 
and contexts (see e.g., Lachance & Confrey, 2003; Krainer, 2006) are addressed at 
least implicitly: 

- Contents are needed that are relevant for all people who are involved (e.g., 
contents that are interesting mathematical activities for the students, challenging 
experiments, observations and reflections for teachers, constructive initiatives 
and discussions for mathematics departments at schools); 

- Communities (including small teams, communities of practice and loosely- 
coupled networks) are needed that allow people to collaborate with each other 
in order to learn autonomously but also to support others' and the whole 
system's content-related learning; 

- Contexts (within a professional development program, at teachers' schools, in 
their school district, etc.) are needed that provide conducive general conditions 
(resources, structures, commitment, etc.). 

The "community" aspect does not mean that initiatives by individual teachers 
are not essential. In contrast, the single teacher is of great importance, in particular 


at the classroom level. However, in order to bring about change at the level of a 
whole department, school, district or nation, thinking only in terms of individual 
teachers is not sufficient ("one swallow does not make a summer"). Research on 
"successful" schools shows that such schools are more likely to have teachers who 
have continual substantive interactions (Little, 1982) or that inter-staff relations are 
seen as an important dimension of school quality (Reynolds et al., 2002). The latter 
study illustrates, among others, examples of potentially useful practices, of which 
the first (illustrated by a US researcher who reflects on observations in other 
countries) relates to teacher collaboration and community building (p. 281): 

Seeing excellent instruction in an Asian context, one can appreciate the 
lesson, but also understand that the lesson did not arrive magically. It was 
planned, often in conjunction with an entire grade-level-team (or, for a first- 
year teacher, with a master teacher) in the teachers' shared office and work 
area. [Referring to observed schools in Norway, Taiwan and Hong Kong: ...] 
if one wants more thoughtful, more collaborative instruction, we need to 
structure our schools so that teachers have the time and a place to plan, share 
and think. 

This example underlines the interconnectedness of the three dimensions content, 
community and context. Most research papers in mathematics teacher education 
put a major focus on the content dimension, much less attention is paid to the other 
two dimensions. In order to fill this gap, the major focus and specific feature of this 
volume is the emphasis on the "community" dimension. This dimension raises the 
"who" question and asks where and how teachers participate and collaborate in 
teacher education. It puts the participants and their ways of participation into the 

How can a volume of a handbook that focuses on participants in teacher 
education deal with this diversity and offer a viable structure? One option is to pick 
one of the manifold differences (formal versus informal; single events versus 
continuous and sustained programmes etc., see above). A very relevant difference 
is the number of participants as, for example, shown (see Table 1) in the case of 
supporting practising teachers to improve their practice in mathematics classrooms 
and related research on their professional growth. Of course, entities like teams, 
communities, networks, schools, districts, regions and nations are only assigned 
provisionally to one of the levels. There might, for example, be teams with ten or 
more people or small schools (e.g., in rural primary schools) with less than ten 
people that teach mathematics (and also most other subjects). 

Table I. Levels of teacher education 

Number of 
M teachers 

Relevant environments (in addition 
to mathematics teacher educators) 

Major mathematics education 
research focus on \. . . ] 

Micro level 


Students, Parents, . . . 

Individual teachers, Teams 

Meso level 


Colleagues, Leaders, ... 

Communities, Networks, Schools 

Macro level 


Superintendents, Policy makers, ... 

Districts, Regions, Nations 


However, in principle, it makes a difference whether we regard the professional 
growth of one or a few teachers in a mathematics department (micro level), or of 
tens of mathematics teachers at a larger school (meso level) or of hundreds or even 
thousands of mathematics teachers in a district or a whole nation (macro level). 

Concerning the three levels, quite different people are interested in the impacts 
of teacher education initiatives: in the case of single classrooms, the students and 
their parents are the most concerned environments; in contrast, superintendents and 
(above all) policy makers are more interested to get a whole picture over all 
classrooms in a country. For example, PISA plays a major role for nations' system 
monitoring of mathematics teaching, but not so much for individual teachers and 
parents. They are more interested in the learning progress of their own students. 
Schools as organizations or networks of dedicated teachers lay somewhat in 
between. On the one hand, a school is important for teachers and parents since this 
organizational entity forms a crucial basis and environment for students' learning; 
for example, this includes important feelings of being accepted, autonomous, 
cognitively supported, a member of a community, safe, taken serious etc. On the 
other hand, reformers need to see schools as units of educational change since they 
cannot reach teachers and students directly. All in all, each of the three levels is 
important and the three should be regarded as closely interconnected. 

However, our knowledge on teachers' learning is not equally distributed among 
these three levels. A survey by Adler, Ball, Krainer, Lin, and Novotna (2005) of 
recent research in mathematics teacher education culminated in three claims: 

- Claim 1: Small-scale qualitative research predominates. Most studies 
investigate the beliefs, knowledge, or practice (and often also the professional 
growth) of individual or of a few teachers. So the focus is primarily laid on the 
micro (and partially on the meso) level, more emphasis is needed on the meso 
and the macro level. 

- Claim 2: Most teacher education research is conducted by teacher educators 
studying the teachers with whom they are working. Also here, the focus is more 
on the micro (and partially on the meso) level since the context of the research is 
prospective or practising teachers' classrooms. Again, more research is needed 
on the meso and the macro level. 

- Claim 3: Research in countries where English is the national language 
dominates the literature. Therefore, we do not know enough about research 
projects in countries where the dominant language is not English. It is more 
likely to get studies from individual researchers from these countries than 
reports about national reforms. There is a big need for comparison of cases of 
reform initiatives (in international publications). 

Since the recent research focus in mathematics teacher education is still on the 
macro level - with a tendency to spread more and more also to the meso level, 
most chapters in this volume also deal with these two levels. This makes it 
necessary to offer a framework that allows us to differentiate between different 
kinds of groups in which teachers participate. A definition by Allee (2000), which 


distinguishes between the notions "teams", "communities" and "networks", is 
helpful here: 

- Teams (and project groups) are regarded as mostly selected by the management, 
have pre-determined goals and therefore have rather tight and . formal 
connections within the team. 

- Communities are regarded as self-selecting, their members negotiating goals and 
tasks. People participate because they personally identify with the topic. 

- Networks are loose and informal because there is no joint enterprise that holds 
them together. Their primary purpose is to collect and pass along information. 
Relationships are always shifting and changing as people have the need to 

I regard these distinctions (of course, others might be possible, too) as crucial 
since, for example, it makes a difference whether initiatives in teacher education 
are predominantly planned from top-down (e.g., by installing teams, task forces 
etc.) or whether they support bottom-up-approaches (e.g., by funding networks of 
teachers that establish their own plans and actions); and, of course, there are 
several approaches in between. The extent to which autonomy is given to 
participants is a crucial social aspect of the community dimension. Of great 
importance is also the extent to which participants feel that they are supported in 
their growth of a competent mathematics teacher and to which extent they belong 
to and might have an influence on a teacher education initiative. This is the kernel 
of the self-regulation theory of Deci and Ryan (2002) which proposes that 
perceived support of basic psychological needs (support of autonomy, support of 
competence and social relatedness) are associated with intrinsic motivation or self- 
determined forms of extrinsic motivation. Since I have become increasingly 
familiar with that approach, which seems to be particularly worthwhile for use in 
teacher education research, only in the last few months it was not possible for this 
volume to take this theory more into account when writing the chapters. However, 
the reader might focus on these ideas when reading the texts. 

The differentiation between teams, communities and networks is used to 
structure this volume. In addition, where appropriate, different chapters focus on 
prospective or practising teachers specifically. In two cases, also a distinction is 
made whether teacher education is held (primarily) in face-to-face or in virtual 
settings. The idea behind working with these differences is that this format could 
facilitate the reader making comparisons between the various strands. Each of 
these chapters gives an overview of our recent knowledge on this particular issue 
and illustrates it through one or a few examples. 


In the remainder of this chapter, each of the six sections and sixteen chapters will 
be introduced very briefly. Since both chapters of the critical respondents in 
Section 6 refer to all texts in the former five sections - and thus necessarily also 
provide descriptions and views on these chapters - I restrict myself here to sketch 


the main topic and to pick out a few issues that could focus readers' attention to 
potentially interesting commonalities and differences. 

Section 1 is on individual teachers as learners. Whereas the chapter by Helia 
Oliveirq and Markku S. Hannula focuses on prospective mathematics teachers, 
Marie-Jeanne Perrin-Glorian, Lucie DeBlois and Aline Robert put practising 
mathematics teachers in the foreground. In both chapters, although focusing on 
individual teachers, not only the importance of the individual but also that of the 
social aspect of teacher learning is highlighted. Similarly, both chapters stress that 
teachers' beliefs, knowledge and practices cannot be separated (e.g., when aiming 
to understand teachers' practice or growth). This makes sense since the focus on 
the individual (and thus on the micro level of teacher education), in general, means 
closer proximity to teachers' practice than in the case of studies at the macro level; 
the nearness and the smaller sample provides the opportunity to go deeper with 
various qualitative methods and thus to see more commonalities and differences 
between teachers' beliefs, knowledge and practices. In contrast, at the meso and (in 
particular at) the macro level, in order to gather a larger data base, it is not easy to 
focus on teachers' beliefs, knowledge and practices within one study and also to 
combine them. In many cases (e.g., where quantitative approaches with larger 
numbers of questionnaires are taken and this is most easily done with regard to 
beliefs), teachers' beliefs (and sometimes also, or instead, their knowledge is 
tested) are measured. 

Section 2 puts an emphasis on teams of mathematics teachers as learners. The 
chapter by Roza Leikin focuses on prospective mathematics teachers. In addition to 
that, Susan D. Nickerson deals with teams of practising teachers. The chapters on 
prospective teachers in Section I and Section 2 have in common that management 
skills and organizational issues are not given special attention. In contrast, both 
chapters on practising teachers deal intensively with issues to do with school 
context. There are a few reasons for this difference. For example, prospective 
teachers are only in schools for relatively short periods during their practicum and 
hence the full impact of school contextual issues might not be evident. Practising 
teachers are in a better position to both identify school contextual issues and to 
address these issues over an extended period. The four chapters in these two 
sections all have in common that the complexity of (mathematics) teaching is 
stressed in various ways. This leads to notions like (teaching) challenges, 
dilemmas, ethical questions, paradoxes, problems, tensions etc. I assume that this 
phenomenon (which cannot be found as much in the other sections) has to do with 
the specific focus of the micro dimension of teacher education. The chapters 
support the view that the complexity of teaching can be seen (at least) as a function 
of the diversity and richness of the mathematics (content), the varying relationship 
between the individual and the groups in which s/he is involved (community) and 
the resources and general conditions (context), which, however, play a bigger role 
in the case of practising teachers than prospective ones. 

Section 3 puts an emphasis on communities and networks of mathematics 
teachers as learners. Fou-Lai Lin and Joao Pedro da Ponte focus on face-to-face 
learning communities of prospective mathematics teachers, whereas Stephen 


Lerman and Stefan Zehetmeier do the same for practising mathematics teachers. In 
contrast to that, Salvador Llinares and Federica Olivero and Marcelo C. Borba 
and George Gadanidis deal with virtual communities and networks of prospective 
and practising mathematics teachers, respectively. Although the four chapters 
represent four different domains where communities (and partially also networks) 
can be initiated or emerge in a self-organized way, they have in common that each 
of them contains various communities representing a diversity of forms, goals and 
purposes. In all four cases, in particular in the face-to-face communities, the 
question of (internal and/or external) expertise or leadership (mathematical 
competent prospective teachers, teacher leaders, principals, qualified experts, more 
knowledgeable others, facilitators, steering group) is raised. This seems to 
highlight that the self-selective nature of communities and networks and the 
corresponding negotiation of goals and activities does not exclude issues of 
expertise or leadership, but in contrast, brings them to the fore. Although in all four 
cases - due to the issue of communities and networks - an important focus is on 
the social aspect, in particular the contents (and to some extent also the contexts) 
are in most cases well described and reflected. It is interesting that also in the 
chapters about virtual communities the integration of face-to-face periods (same 
room, same time) or at least synchronous periods of online-participation (same 
time, e.g., working interactively on the same mathematical problem using the same 
software) are suggested and realized. On the other hand, the flexibility concerning 
time and thus autonomy that asynchronous forms of online teacher education give 
to (in particular practising) teachers is so attractive that about 95% of continuing 
teacher education programmes of a particular university are organized online. One 
major difference between face-to-face and virtual learning environments seems to 
be generated by the tools: new technologies and tools seem to change the form of 
communication. Partially, the tools are not only regarded as mediators but also as 
co-actors in the communication process. The new technical options in the web 
allow a transformation from read-only to read-and- write communication (e.g., 
weblogs and wikis). Probably, it is not by chance that the chapters and many 
references on virtual forms of teacher education stem from countries with large 
distances to cover like Australia, Brazil, Canada, or Spain. It is worth noticing that 
beginning with Section 3 (and not in the sections before that which primarily 
belong to the micro level) the question of sustainability of teacher education is 
raised in several chapters. 

Section 4 shows a shift of focus to the development of schools, regions and 
nations as a means of improving mathematics teaching and learning. Elham 
Kazemi puts an emphasis on school development and thus relates to the meso level 
most extensively. Paul Cobb and Thomas Smith deal with district development as a 
means of improving mathematics teaching and learning at scale, the same focus is 
taken by John Pegg and Konrad Krainer regarding regional and national reform 
initiatives. These two chapters put their primary focus on the macro level, the 
former one more extensively on the theoretical foundation and on planned 
activities, the latter more on reform initiatives (carried out or still continuing) 
reflecting and comparing cases from four countries concerning their genesis, goals, 


results of evaluation and research etc. Entering the meso and macro level, thus 
focusing on the improvement of a larger number of mathematics classrooms, the 
organizational aspects of change (development of schools, districts etc.), new 
relevant environments for mathematics teaching and learning like parents, 
principals, other kinds of leaders, but also organizations like universities and 
ministries come into play. They are regarded as crucial co-players of bringing 
about change, although the teachers and the students (and their interaction) are seen 
as the key for systemic change. Given the complexity of the different actors (from 
individuals, teams, communities, networks, organizations, media etc.), places, 
resources, goals to be defined or negotiated, reflection and evaluation, ways of 
communication and decision making, the content - at least at a first view - 
becomes less important. However, it is naive to assume that bringing about change 
in mathematics teaching at a large scale can be reduced to defining central 
regulations, working out national mathematics standards, writing textbooks and 
research papers, golden rules about good teaching and to transfer it to individual 
teachers. Thinking that way, the autonomous culture established over years at 
schools, districts etc. are underestimated and not taken seriously. Systemic change 
needs to take into account the participants of initiatives at the micro, the meso and 
the macro level in order to avoid "system resistance" at one or more of these levels. 
The three chapters have in common that they deal with notions like (inter)national 
assessments and benchmarks, brokers, change agents, differences between 
designed and lived organizations, economic development, intervention strategies, 
key boundaries objects, leadership content knowledge, regional networks, 
stakeholders, support structures, systemic reform etc. which indicate that 
interconnections between different levels, social systems etc. need to be balanced. 
On the one hand, it is understandable that the three chapters deal to a large extent 
with practising teachers. On the other hand, the lack of putting an emphasis on 
prospective teachers might reveal the problem that schools and districts usually do 
not have strategies for integrating novice teachers in a way so that both sides can 
profit (or even do not have strategies for personal development at all). The journey 
from the micro to the macro level shows that not only the focus of attention is 
shifting from one class or a few to maybe thousands, but that also larger entities of 
teachers and their professional growth are investigated, using more qualitative 
studies at the micro and more quantitative studies at the macro level. Of course, the 
participants of teacher education and professional development initiatives (and not 
the students) form the focus of this volume, nevertheless, there is a tendency to 
look at students' growth (affective or cognitive) rather at the macro level, in 
particular with regard to (inter)national assessments that often are the starting point 
of national reform initiatives. 

Section 5 focuses on teachers and teacher educators as key players in the further 
development of the mathematics teaching profession. Gertraud Benke, Alena 
Hospesovd and Marie Tichd analyse the use of action research in mathematics 
teacher education. Barbara Jaworski discusses a specific way of collaborating 
between teachers and didacticians. Finally, Nanette Seago reflects on the 
mathematics teaching profession in general. These three chapters form a kind of 



reflection on teacher education that does not go along the levels introduced in this 
volume, but along particular thematic issues relevant to the participants and 
organizers of teacher education. Often underestimated, however of particular 
importance, are forms of teacher education where prospective or practising 
teachers are investigating their own practice. Therefore, (critical) reflection is a key 
notion in the first of these chapters. It is interesting to follow the different ways 
action research is used and the issues educational researchers have to attend to 
when supporting teachers engaged in such projects. This is the bridge to the second 
chapter in this section where the focus is on the collaboration between teacher 
educators (didacticians) and practising teachers as partners and co-researchers in 
inquiry communities. Whereas the first chapter combines general considerations on 
characteristics and aspects of action research with examples from two countries, 
the second chapter explains the theoretical background of building and sustaining 
environments for co-learning inquiry and demonstrates this by giving an insight 
into a specific project, illuminating its issues and tensions. The third chapter 
addresses the professional ization of mathematics teaching, the specialized 
knowledge mathematics teachers need to have and design principles for 
professional development. This is illustrated by a specific mathematics teacher 
development programme that uses video-records of classroom practice. Like all 
other chapters, these three chapters aim at addressing general insights and research 
results as well as particular views on a few cases in order to make the general 
aspects more concrete and authentic. 

Section 6 contains two chapters written by Gilah C. Leder and Heinz Steinbring 
who look in a kind of "critical response" to the whole volume from two specific 
perspectives. The first chapter follows the structure of this volume, refers to all 
sections, describes the chapters in the order they are published, sifts out interesting 
issues and indicates the complexity and diversity of the field and the variety of 
contributions, approaches, theoretical and practical stances in this volume. In 
contrast, the last chapter offers a theoretical perspective on fundamental problems 
in the context of investigating the communication issues raised in this volume. In 
particular, the theory-practice-problem within mathematics education and the 
importance of the subject (mathematics) and of teaching activity are raised. Doing 
that, links to all previous sections and chapters are made. These two chapters form 
a reflective closure of the whole volume. 


Together with Terry Wood, I wish to gratefully thank all authors of this volume for 
their great efforts and efficient work on this challenging joint endeavour of putting 
the participants and community dimension into the foreground. We appreciate very 
much that Gilah C. Leder and Heinz Steinbring not only wrote the final chapters 
but also reviewed all chapters. In particular, we also want to highlight that the 
whole review process (each chapter was also peer-reviewed by a first author or a 
colleague of another chapter) was done in a critical but always constructive way. 
Furthermore, I thank Dagmar Zois for her helpful correction and layout work 


concerning all chapters. We hope that you, as the readers of this volume, will gain 
new and interesting insights into various aspects of mathematics teacher education 
that partially gorge new pathways; they are not paved, they are tentative in nature 
and work in progress. You are invited to collaborate and to be a participant in this 
joint process. 


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Researching mathematics teacher education. Educational Studies in Mathematics, 60, 359-38 1 . 
Allee, V. (2000). Knowledge networks and communities of practice. OD Practitioner, Journal of the 

Organization Development Network, 32(4), 4-13. 
Deci, E. L., & Ryan, R. M. (Eds.). (2002). Handbook on self-determination research Rochester: 

University of Rochester Press. 
Krainer, K. (2006). How can schools put mathematics in their centre? Improvement = content + 

community + context. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlkova (Eds.), Proceedings of 

the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 

1 , pp. 84-89). Prague, Czech Republic: Charles University. 
Lachance, A., & Confrey, J. (2003). Interconnecting content and community: A qualitative study of 

secondary mathematics teachers . Journal of Mathematics Teacher Education, 6, 107-137. 
Little, J. W. (1982). Norms of collegiality and experimentation: Workplace conditions of school 

success. American Education Research Journal, 19, 325-340. 
Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In 

A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education. 

Past, present and future (pp. 429-459). Rotterdam, the Netherlands: Sense Publishers. 
Reynolds, D., Creemers, B., Stringfield, S., Teddlie, C, & Schaffer, G. (Eds.). (2002). World class 

schools. International perspectives on school effectiveness. London: Routledge Falmer. 
Sowder, J. (2007). The Mathematical Education and Development of Teachers. In F. Lester (Ed), 

Second handbook of research on mathematics teaching and learning (pp. 1 57-223) Greenwich, CT: 


Konrad Krainer 

Institut fur Unterrichts- und Schulentwicklung 

University of Klagenfurt 







Studies on Their Professional Growth 

There are two types of goals for those who learn to become mathematics teachers. 
Firstly, they need to learn mathematics, and secondly, they need to learn how to 
teach it. On the one hand, those who specialize in order to become secondary level 
subject teachers usually have a strong mathematical background but they may 
have weaker identities as teachers. Those who become elementary education 
teachers, on the other hand often have problems with the mathematical content 
themselves, but they have stronger identities as teachers. In this chapter we will 
take three perspectives to mathematics teachers' learning during their teacher 
education: 1) the development of teachers ' knowledge and beliefs during that 
period; 2) the development of necessary skills for the teaching profession, such as, 
observing and interpreting classroom incidents, and reflecting on them, as well as 
different modes of interaction with the class; and, 3) the adoption of a productive 
disposition and identity as a teacher who is a reflective and collaborative 
professional and is willing to engage in future professional development. The last 
one will be illustrated with a case study focusing on the development of the identity 
of beginning secondary mathematics teachers. We then discuss the challenges to 
initial teacher education of a perspective centred on the idea of the individual 
prospective teacher as learner. 


When we speak about becoming a (mathematics) teacher, we are talking about a 
learning experience that can be seen as an individual one, but also as a social one. 
Our focus in this chapter is on research on teacher education, mainly work 
published over the last ten years, which centres on the development of the 
individual prospective mathematics teacher and embraces both the uniformity and 
the diversity of individuals in this process. This would seem to be an easy task 
since for many years studies have centred on a small number of participants and 
focused on the individual. Adler, Ball, Krainer, Lin, and Novotna (2005) provided 
a detailed table which revealed that out of the 160 studies on teachers that were 
reported in the proceedings for the Psychology of Mathematics Education (PME), 
Journal of Mathematics Teacher Education (JMTE) and Journal of Research in 
Mathematics Education (JRME) over the years (1999-2003), 21 were reports on a 
single teacher, while only 10 had one hundred or more participants. In the review 
of research presented in PME about teacher education, from 1998 to 2003, the 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 13-34. 
© 2008 Sense Publishers. AH rights reserved. 


majority of studies also had less than 20 participants (Llinares & Kramer, 2006). 
Nevertheless, concentrating on the learning of the individual prospective teacher 
proved to be a very complex task because there are many studies but these quite 
often do not target the individual with great depth. 

Since the launch of JMTE in 1998, this journal is the main forum for publishing 
studies on mathematics teacher education, and also in our review this journal has 
been a very important source. We also searched for papers in Educational Studies 
in Mathematics (ESM), JRME and PME conference proceedings, mainly between 
1998 and 2007. We also included a few studies from other publications that 
seemed particularly relevant for the purposes of this chapter. 

When we study prospective teachers' learning to become a teacher, we asked, 
what kind of learning are we expecting to see? To attain our goal, we have 
identified three perspectives on learning that we shall use to structure this chapter: 
1 ) one view perceives learning as acquiring knowledge or beliefs, 2) another view 
perceives learning as mastering a skill, for example, the ability to observe and 
reflect, and 3) the third perspective perceives learning as adoption of a certain 
disposition, for example, an identity or an orientation. 


For the last two decades, research on prospective teachers' knowledge and beliefs 
has received enormous attention from the community of mathematics teacher 
educators. The recent work of Llinares and Krainer (2006), and Ponte and 
Chapman (in press) provide an extensive review of the literature on these areas and 
offer a good overview of the themes that are being addressed and of the knowledge 
that has been accumulated. In the present chapter we want to stress the individual 
dimension of learning to teach as the development of beliefs and knowledge. 

Prospective Teachers ' Beliefs 

Research on teacher's beliefs (see also Leikin, this volume) has a long tradition in 
research on teacher education and has been "perhaps the predominant orientation in 
research on teachers and teacher education" (Lerman, 2001, p. 35). Changes in 
beliefs are assumed to reflect development. Experiences and reflection are two 
basic sources of influence that are considered to be important in the formation, 
development and change of beliefs. Llinares and Krainer (2006) highlight the 
importance of early experiences as a learner of mathematics in the formation of 
teachers' beliefs. Once beliefs have been formed, they are not easy to change. 
Liljedahl, Rolka, and Rosken (2007) have identified three different methods used 
to change elementary education teachers' mathematics-related beliefs. The first 
method is to challenge their beliefs. Many beliefs are held implicitly, but when 
they are challenged, they can become explicit and subject to reflection, creating the 
opportunity for change. The second method is to involve them as learners of 
mathematics, usually in a constructivist setting. A third method for producing 
changes in belief structures is to provide prospective teachers with experiences of 



mathematical discovery, which seems to have a profound, and immediate, 
transformative effect on their beliefs regarding the nature of mathematics, as well 
as the teaching and learning of mathematics (Liljedahl, 2005). Llinares and Kramer 
(2006) note that it is often assumed, not always correctly, that learning 
mathematics through inquiry will influence prospective teachers' beliefs and 
attitudes in a positive way, namely in regarding the implementation of those 
methodologies in their practice as in the case described by Langford and Huntley 

In some cases it seems that the use of "reform-oriented" teaching material in 
teacher education has been useful in influencing prospective teachers' beliefs about 
the nature of mathematics. In Lloyd (2006, p. 77), one prospective teacher 
explained: "I am learning about how to look for reasons and explanations as 
opposed to simply believing 'the rules' that some really ancient dead guy came up 
with. I prefer being able to use my own mind in solving problems". However, 
according to Szydlik, Szydlik, and Benson (2003, p. 254), research has shown that 
prospective teachers tend to "see mathematics as an authoritarian discipline, and 
that they believe that doing mathematics means applying memorized formulas and 
procedures to textbook exercises". 

One of the big issues in research on teachers' beliefs is the mismatch between 
teachers' personal theories of learning and their actual teaching practices that seem 
not to support the theory. This is a special case of a more general mismatch 
between espoused and enacted beliefs. It has been suggested that theory and 
practice can be connected if integrated in authentic contexts. Bobis and Aldridge 
(2002) have designed a continuum of authentic contexts for their master level 
elementary teacher education course. Their students experienced mathematics 
teaching and reflected on their experiences in typical student workshops and 
practice schools, but they also used university based clinics and school-based small 
group teaching. Somewhat paradoxically, they only were able to report changed 
beliefs of their students during the course, as they have not followed the students 
after they moved to their positions as teachers. The possibility of a dynamic 
interaction between coursework and fieldwork in the development of alternative 
perspectives on teaching and learning is illustrated by Ebby (2000). This research 
shows that the three prospective teachers learned from their teacher education 
coursework and fieldwork in quite different ways, influenced by their beliefs, 
dispositions, and individual experiences. 

In many studies, the development of prospective elementary education teachers' 
learning of content knowledge is intertwined with their changing beliefs about the 
nature of mathematics (e.g., Amato, 2006; Nicol, Gooya, & Martin, 2002; Olsen, 
Colasanti, & Trujillo, 2006). One example of change in beliefs as a result of 
learning content is the case of the prospective elementary education teacher, 
Jennifer (in Davis & McGoven, 2001), in the context of a mathematics content 
course that was built around problem solving and included class discussions and 
reflective writing. Jennifer begun the course holding a typical belief that 
mathematics is about applying the right formula to get the right answer. She 
learned to see and value the mathematical connections between different tasks and 



her test performance improved from not satisfactory to excellent. One 
characteristic of Jennifer was her extensive and elaborated reflective writing. A 
similar bond between content knowledge and beliefs was identified by Kinach 
(2002), when she was challenging her prospective secondary mathematics teachers 
to provide instructional explanations for subtraction of negative numbers. She and 
her prospective teachers found themselves "to be in disagreement about what 
counted as an explanation, and even more fundamentally, about what counted as 
knowing, and therefore learning mathematics" (Kinach, 2002, p. 179). Hence, the 
problem of teaching mathematical content was essentially also a problem of 
changing the prospective teachers' beliefs about the nature of mathematics. 

Prospective Teachers ' Knowledge 

The above indicated interdependence of knowledge and beliefs in learning to teach 
becomes more explicit from the growing body of empirical research dealing with 
both concepts (e.g., Llinares, 2003) and how theory is being built around them 
(Pehkonen & Pietila, 2003). However, the analysis of the knowledge that 
prospective teachers should develop to teach is part of a long enterprise in research 
on teacher education (Ponte & Chapman, in press). 

It is well recognized that research on teachers' knowledge is strongly influenced 
by the work of Shulman (1987), focusing mainly on the mathematical knowledge 
of teachers and on pedagogical content knowledge. Regarding the first aspect, 
studies have confirmed prospective teacher misconceptions (or lack of conceptual 
understanding) in different branches of subject matter knowledge (Llinares & 
Krainer, 2006; Ponte & Chapman, in press). This problem is more frequently 
addressed among prospective elementary education teachers, but also prospective 
secondary mathematics teachers at times struggle to attain a deep understanding of 
mathematics (Kinach, 2002). As one prospective teacher wrote in her journal: 
"Over the past semester, there have been a few times where 1 have stopped and 
realized that there are some math concepts that 1 thought 1 knew, but actually 
didn't" (Kinach, 2002, p. 176). This reflection about what one knows and how one 
is learning is fundamental in the process of becoming a mathematics teacher. In 
many countries, secondary mathematics teachers have a solid mathematic 
education, with three or more years of mathematics courses in several domains. 
However, the failure in some courses is very high, even for prospective teachers 
who see themselves as good at mathematics in secondary school. One of the 
reasons for this may be the way mathematics is taught to prospective teachers, and 
not only because of the content to which they are exposed. 

One important question that arises concerning individual prospective teachers as 
learners is how they evaluate the mathematical knowledge they are developing in 
terms of the development of subject knowledge for teaching. For example, some of 
the beginning mathematics teachers studied by Oliveira (2004) did not see much 
relation between the mathematics they learnt in the mathematics courses they 
attended for three years and the way they are to teach. But they still considered that 
the study of "hard mathematics" enhanced their mathematical reasoning, giving 



them confidence to answer their students' questions and to solve mathematical 
problems. Curiously, the beginning teacher who scored the highest marks in the 
mathematics courses, and who deliberated between becoming a secondary 
mathematics or a mathematician, was the one who recognized that sometimes he 
had great difficulties in solving the mathematical problems given in methods 
courses. This made him question the "hard mathematics" he had been studying 
during that period in mathematics courses. 

Certainly, prospective mathematics teachers also develop mathematical 
knowledge in different ways, as a consequence of their past experiences with 
school mathematics and in higher education. Sanchez and Llinares (2003, p. 21) 
describe four university graduates, prospective secondary mathematics teachers, 
who expressed different ways of knowing the function concept; this, according to 
the authors, "influenced what they considered important for the learner and 
affected their use of the modes of representation in teaching that were considered 
as teachers' tools to obtain his/her teaching goals". This raises another important 
question concerning prospective teachers' subject matter knowledge and its 
relationship with their emerging pedagogical content knowledge. Tirosh (2000, p. 
329) argued that teacher education needs to take into account the prospective 
teachers' knowledge of students' common responses to given tasks, considering 
that "such knowledge is strongly related to prospective teachers' SMK [subject 
matter knowledge]". From a situated perspective on learning, the research 
developed by Llinares (2003, p. 205) showed that the case analysis that prospective 
teachers developed in interaction provided a context for them to attempt to 
understand the students' way of thinking and simultaneously "to think about the 
meanings of some elements of their subject matter knowledge and pedagogical 
content knowledge". With respect to the process of learning in a small group of 
four prospective primary teachers as learning in a community of practice, the 
author explained that they engaged differently in the task and assumed different 
responsibilities in exploring new domains. Despite their individual engagement, the 
interaction among them was very important for the reification of beliefs, something 
that Llinares contends could not be attained if reflection would stand merely at a 
discursive level. 


The studies reviewed provide evidence, to a certain extent, of the attention paid to 
the individual prospective teacher. Collectively they show that there are important 
differences that can be relevant influences on how they are going to experience 
teacher education and develop as prospective teachers. Therefore, those differences 
should be taken into account, when designing and developing teacher education 
programmes. For teacher educators this means to further develop two competences: 
to notice such differences (see e.g., Krainer, 2005, referring to Willke, 1999, who 
writes about the art of precise observation being typical for experts in contrast to 
laymen); and to produce relevant differences (which Willke, 1999, in Krainer, 
2005, defines as interventions). Such an intervention in teacher education could 



be, for example, to make differences among prospective teachers' views fruitful for 
their joint reflection on these differences. 


The practicum experience is generally considered to be an integral part of teacher 
preparation. However, it has been argued that prospective teachers need to develop 
a critical stance towards their field experiences and a "certain distance" from the 
ongoing actions (Jaworski & Gellert, 2003) in order to reflect about the events of 
the classroom, otherwise "field experiences perpetuate apprenticeship and trial-and 
error views of teaching" (Mewborn, 1999, p. 318), and oversimplify its nature. 

In this section, we are looking at the role of practice in learning to teach from 
three different - although in some cases complementary - approaches: 1 ) observing 
teachers and students engaged in mathematical activity, 2) prospective teachers' 
interaction with students, and 3) prospective teachers' reflection. In our opinion, 
these three ways of seeing how prospective teachers learn to teach entail different 
activities that are nuclear for the teaching profession (see Seago, this volume). 
There are additional important issues regarding the PCK prospective teachers are 
developing, namely when planning and selecting appropriate tasks and resources 
for students. 

Observing Teachers and Students Engaged in Mathematical Activity 

When prospective teachers come in contact with pedagogical situations in 
elementary and secondary schools, they have already spent many hours in 
classrooms as students. Therefore, the classroom environment and pedagogical 
situations may seem quite familiar for them. It is necessary for them to produce a 
shift of perspective from one of a prospective teacher to that of a teacher: "teaching 
is fundamentally about attention; producing shifts in the locus, focus, and structure 
of attention, and these can be enhanced for others by working on one's own 
awareness" (Mason, 1998, p. 244). It is not surprising that novice and expert 
teachers have different competencies in observing mathematics video taped 
teaching episodes. In a study that used eye-tracking technology, it was found that 
prospective elementary teachers attended more to mathematics content, while 
experienced teachers and mathematics educators focused more on the activities of 
the teacher and student (Philipp & Sowder, 2002). Some teacher education 
programmes prepare prospective teachers to conduct clinical interviews, 
considering that those are useful tools for learning to understand how students 
think mathematically and to "develop an increased awareness of the ways in which 
people learn mathematics" (Schorr, 2001, p. 159). Ambrose (2004) argues that by 
having prospective elementary teachers interviewing children, namely on specific 
difficult mathematical concepts, they realise that the mathematics they were 
supposed to teach was not so simply and required conceptual understanding. 

Morris (2006) focused on the prospective teachers' ability to collect evidence 
about student learning to analyse the effects of instruction and to use the analysis to 



revise instruction. The results of this study suggested that prospective teachers 
possessed some initial diagnostic and revision skills, and that the quality of 
analysis was influenced by the instruction given prior to the task, namely that when 
a lesson was perceived as problematic it encouraged them to look more closely at 
students and to ask themselves questions about the situation. 

In the case of a programme that used a multimedia resource, elementary and 
secondary prospective teachers investigated assessment and teaching strategies in a 
situated learning environment (Herrington, Herrington, Sparrow, & Oliver, 1998). 
They could observe video clips of teachers who were using different assessment 
and teaching strategies, and at the same time they had access to teachers' 
discussions about their approaches. Prospective teachers recognized the influence 
of this environment in the assessment strategies they adopted later in their teaching 

These initial experiences with the classroom reality and with students can have 
an impact on the development of certain attitudes and knowledge that are central 
for teaching, namely a disposition to listen to students and to try to understand how 
they think and reason mathematically. In all these studies, prospective teachers' 
practice is very limited in time and scope, but these experiences constitute also an 
opportunity for the prospective teacher to start feeling what it is like to be a 
teacher. Assuming that the teacher's role in classroom is much more complex than 
working with one student or a small group of students on a specific theme or 
concept, and that the teaching situations are so diverse, multimedia explorations 
can assist prospective teachers in grasping that reality. 

Prospective Teachers ' Interaction with Students 

In spite of being, by definition, an interactive profession, teaching through 
meaningful interactions with students is one of the most demanding aspects of that 
practice. Research shows that prospective teachers often tend to focus their 
attention on issues concerning class management and pedagogy (Van Zoest & 
Bohl, 2002), considering subject matter learning less problematic. 

Moyer and Milewicz's (2002) study on prospective teachers' questioning skills 
looked at the developing interaction skills of practicing teachers. They argued, that 
"when open-ended questioning is used and there are many right answers, the 
learning environment becomes complex and less predictable as teachers attempt to 
interpret and understand children's responses" (p. 296). In their research they gave 
prospective elementary teachers the task to interview individual students on 
mathematical concepts and later to reflect on the audio-recorded interview. They 
concluded that the prospective teachers did not yet have good questioning skills, 
but that the experience of interviewing "is a first step towards developing the 
questioning strategies that will be used in the multi-dimensional, simultaneous, 
unpredictable environment of the classroom" (p. 311). 

The nature of discourse in the mathematics classroom is another way of looking 
at teachers' learning. Blanton (2002) describes the experiences of eleven secondary 
mathematics teachers in a course that attempted to challenge their notions about 



discourse as univocal or dialogical. Throughout the course, prospective teachers 
developed a positive stance toward dialogic discourse but they felt there were 
many obstacles to developing that kind of discourse in their practice. They 
recognized that their discourse was predominately univocal as, for example, they 
felt the need to structure and control the class. Blanton suggests that it is difficult 
for prospective teachers to develop a dialogic discourse due to aspects such as: 
sharing authority with students; focusing on students' thinking by acknowledging 
and incorporating their ideas; and in balancing the need for dialogic discourse with 
the time constraints of "covering the curriculum" (p. 149). 

There is evidence from research that many prospective teachers begin their 
practice trying authentically to assume a new role as teacher, one that is attuned 
with reform ideas presented in university courses, but lack the skills to implement 
them. For example, Lloyd (2005) described the internship of one prospective 
secondary mathematics teacher who started his student teaching with an approach 
that was student-centred, relying on group work and the use of manipulatives and 
technology but that left unchanged the mathematical nature and content of 
students' activity. As a consequence of the students' complaint that they "were not 
being taught" (p. 457), he recognized that he had been emphasizing "how to solve 
problems rather than why certain ideas and methods are related" (p. 457). The 
prospective teacher was learning through interaction with students by attending to 
the students' feedback and then reflecting on his own actions as a teacher. 

Questioning students, and communicational aspects, in a more general way, are 
central issues in learning to teach, but research shows that these are quite 
demanding for prospective teachers. Implementing reform ideas or changing 
approaches to mathematics teaching is challenging in many of the diverse 
classroom contexts (see also Leikin, this volume). The professional growth of 
prospective teachers depends on the opportunities to effectively experiment with 
teaching and in a context where they are stimulated to interact with students in 
meaningful ways, with the support of those responsible for their education as 

Teachers ' Reflection 

Reflection has become a popular concept in teacher education and it is assumed to 
be a critical element in belief change in the process of becoming a teacher (Llinares 
& Krainer, 2006; Hannula, Liljedahl, Kaasila, & RSsken, 2007; Liljedahl, Rolka, & 
RSsken, 2007). Elements that have been found to support reflection are 
collaboration (Kaasila, Hannula, Laine, & Pehkonen, 2006) and use of multimedia 
(Masingila & Doerr, 2002; Goffree & Oonk, 2001). 

Reflection is seen as an indispensable element in the process of learning from 
and in practice. Many teacher education programmes have implicitly addressed the 
idea of promoting reflection about prospective teachers' practice by creating 
moments for discussing their classroom lessons with tutors or/and mentors and 
helping them "to analyse in more detail their own teaching practices" (Jaworski & 



Gellert, 2003, p. 843). This is done in different programmes in a more or less 
structured way. 

In a study conducted by Artzt (1999), prospective secondary teachers were 
engaged in structured reflection on their teaching. The course built on prospective 
teachers' existing knowledge and beliefs and used both pre-lesson and post-lesson 
reflective activities. Prospective teachers were encouraged to think about the 
decisions they made in light of their goals for students. This research presents two 
contrasting cases. Mrs. Carol was revealed to be insecure about her teaching and 
mathematical abilities and about becoming a teacher. Through her writing 
assignment the supervisor could detect a low self-esteem. But these feelings 
constituted a motivation for her to be open-minded about learning new approaches 
and she started to plan and implement different lessons. By the end of the semester, 
she had understood the importance of reflecting about her underlying beliefs about 
students and how they learn. In contrast, the other prospective teacher, Mr. Wong, 
had strong beliefs about the "right" way to teach that left him inflexible and 
unmotivated to learn new instructional approaches. So the course did not cater 
equally for all prospective teachers. 

Another study, about the teaching practicum, examines the role of reflection in 
prospective teachers' practice by looking at how they use their pedagogical content 
knowledge in solving problems identified from the classroom (McDuffie, 2004). 
The elementary prospective teachers approached teaching as a problem solving 
endeavour, since their internship included completing a classroom-based action 
research project on their own teaching (see also Benke, Hospesova, & Ticha, this 
volume), and focused on facilitating their understanding and anticipating problems 
in teaching and learning. 

As prospective teachers' observation skills are often limited, new technology 
has been introduced as new means for reflection. Multimedia allows the same 
episode to be watched several times and also to hear interviews of the people 
involved in the episode more than once. In the programme Multimedia Interactive 
Learning Environment (MILE), video clips of actual teaching situations are used, 
and elementary education students are invited into reflective discussions about the 
situations in order to enhance their practical knowledge (Goffree & Oonk, 2001). 
Nevertheless, the authors consider that it is still necessary to understand better 
"how to establish a link between the fieldwork of teacher education students and 
their investigation and discourses in MILE" (p. 143). 

The connections established by prospective teachers between multimedia case 
studies and their practice was the focus of one study presented by Masingila and 
Doerr (2002), involving grade 7-12 prospective mathematics teachers. The cases 
were created in a way "that would reflect the complexities of classroom 
interactions, teacher decisions, and students' mathematical thinking" (p. 244), in 
order to promote investigation, analysis and reflection. Prospective teachers were 
asked to select a specific issue from their own practice that they saw addressed in 
the teacher's practice there and discuss it. They were able to discuss the case, 
taking into account their own practice, and focused on complex issues instead of 



the usual management concerns. However, the authors did not investigate if the 
prospective teachers subsequently changed their teaching practices. 

Mewborn (1999), studying prospective teachers who participated in a field 
experience during a mathematics methods course, claimed that the ways they went 
about making sense of what they observed in a mathematics classroom "can be 
characterized as reflective thinking in the manner described by Dewey" (p. 324). 
This happened when the locus of authority was internal to the prospective teachers, 
and was promoted by the teacher educator and the classroom teacher who tried to 
remove themselves "from positions of authority" (p. 337). They tried not to answer 
directly to prospective teachers' questions, instead they "turn questions back to 
them and encouraged them to rely on their peers for evaluation of their ideas" (p. 

It is not surprising that reflection on teaching practice is such a difficult task for 
prospective teachers since learning in practice is so demanding for them. Even 
more challenging is to reflect about the events and to change rapidly the course of 
the lesson, adapting the plan that they made. Sometimes, they are supposed to 
reflect "on mathematical activity while participating in the discourse" (Lloyd, 
2006, p. 462). This cannot be seen exclusively as a skill or disposition to reflect on 
the moment but as contingent action that relies on the (prospective) teacher's 
confidence and willingness to assume risks (Rowland, Huckstep, & Thwaites, 
2005), in a situation where, at the same time, she or he is under the scrutiny of 


One of the classical problems of teacher education has been the bridging between 
theory and practice. Observing and practice teaching has been used as a general 
solution to this problem. Looking at the studies reported above it is possible to 
recognize the important role of university tutors and teacher mentors in helping 
prospective teachers to observe learning incidents in their complexity, to generate 
fruitful interaction with students, and to reflect upon them, even when that role is 
not specifically targeted. New multimedia learning environments hold a lot of 
promise to develop prospective teachers' understanding about the complexity of 
mathematics teaching, but educational environments that are supportive and 
flexible, taking into account their unicity and individuality, are also very important 
for their professional growth. 


Our last approach to prospective teachers' development through teacher education 
looks at their disposition as a teacher and at what they want to do as professionals 
(and as persons). Often the professional and personal aspects are intertwined and 
the development of a skill is necessary to promote the inclination to use it (Peretz, 
2006). Learning to teach is also a "process of becoming" (Jaworski, 2006, p. 189), 
developing a new identity, one that integrates a professional side. 



Disposition towards Mathematics and Its Learning 

Disposition and identity issues have their specificities in the case of elementary 
teachers, considered as generalist teachers, and in the case of secondary teachers, 
usually regarded as specialized in mathematics. Especially among prospective 
elementary education teachers, there are several whose attitude towards 
mathematics is negative; sometimes they develop even a more serious condition - 
mathematics anxiety. As such, it is important that elementary teacher education 
programmes help students overcome this problem (e.g., Kaasila, 2006; Liljedahl et 
al., 2007; Pietila, 2002; Uusimaki & Nason, 2004). These examples collectively 
use what is often referred to as a therapeutic approach (Hannula et al., 2007). One 
of the elements of a therapeutic approach is the effort to provide students with 
positive experiences with mathematics. This can be achieved within the context of 
elementary mathematics, where hands-on material is used to give prospective 
teachers an example of teaching in a constructivist way, whilst at the same time 
providing an opportunity for many of them to really understand mathematics for 
the first time in their life (Pietila, 2002; Amato, 2006). In a supportive classroom 
climate, anxious prospective teachers may express their thoughts and feelings and 
ask for advice without fear of stigmatisation (Pietila, 2002). However, experience 
alone is not sufficient for a major change in students' mathematical self-concept - 
it needs to be supported by reflection and peer collaboration (Hannula et al., 

On the one hand, a negative view can seriously influence students' becoming 
good mathematics teachers (Uusimaki & Kidman, 2004), on the other hand, 
prospective teachers who have experienced only success in school mathematics 
may find it hard to understand students for whom learning is not so easy (Kaasila, 
2000), and that happens quite often with secondary mathematics teachers. These 
teachers very often identify themselves very strongly with the subject they teach, 
but develop a stronger frame of reference to certain forms of teaching (Sowder, 
2007), which constitutes a challenge to teacher education programmes. In fact, 
during their school years, prospective teachers begin to develop personal beliefs 
about teaching and perspectives about teacher's and student's roles, and the nature 
of the subject they are going to teach, through which they will interpret teacher 
education programmes. 

Assuming Different Roles in Teacher Education 

Another issue in research concerning the development of prospective teachers' 
professional identity is whether they identify themselves as learners of 
mathematics or if they already identify themselves as (prospective) teachers of 
mathematics. Although they are learning to become teachers, at the beginning of 
their studies they are primarily identifying themselves as students. This duality has 
been explicitly expressed, for example, in Bowers and Doerr (2001). Sometimes 
there is a multiplicity of roles to perform as documented in one case study 
presented by Stehlikova (2002). Molly, a prospective secondary mathematics 



teacher, over the course of her five-year education developed from "a role of a 
pupil [who] expected to be taught [...] into an independent problem solver, 
autonomous learner, 'mathematician' at times, teacher and teacher researcher" (p. 
245). Although Molly was an exceptional case of an enthusiastic learner, she 
exemplifies the variety of different roles a prospective teacher is expected to unite 
into a coherent identity. 

Teachers ' Interactions with Students 

Interactions with students constitute one of the main configurative elements in the 
process of identity construction. Quite often prospective teachers meet with 
students who do not match their expectations (Munby, Russell, & Martin, 2001). 
However, these interactions can promote a change in their perspectives. For 
example, Skott (2001) documented that learning to be a mathematics teacher 
involved more than "merely" teaching mathematics, and that it was important to 
(re)evaluate their priorities concerning teaching practice, namely the necessity to 
attend to students' problems. Oliveira (2004) also found that some secondary 
mathematics teachers started to realise, early in their career, that they had an 
important role as educators in spite of having to teach a socially "strong" subject. 

Developing an Identity in Different Contexts 

Sfard and Prusak (2005) define identities as collections of those narratives that are 
reifying, enforceable and significant. According to these authors, different 
identities may emerge in different situations and that might happen with 
prospective teachers as they emerge in different contexts. In a longitudinal study 
with four elementary prospective teachers, Steele (2001) illustrated "some of the 
problems and realities of the workplace that interfere with teachers sustaining a 
change in conceptions" (p. 168). Two of them remained attached to the 
conceptions developed in teacher education while the other two did not. One of 
these prospective teachers felt specially pressured by the school administration and 
implemented a "teacher-proof curriculum" (p. 169). This was also the prospective 
teacher who showed less change in her conceptions during the period of teacher 
education and simultaneously was more confident about mathematics. Steele 
conjectures that "perhaps her past experiences of learning mathematics were in the 
end more influential in her approach to teaching" (p. 168). 

Opportunities for learning during teaching practice are shaped by the 
characteristics of the contexts in which teacher education occurs, in some instances 
by the strong evaluative flavour (Mewborn, 1999; Johnsen Heines & Lode, 2006) 
and they are also fused with "networks of power" (Walshaw, 2004). Using insights 
from the post-structural ideas of Foucault, Walshaw discusses what it means to 
engage in pedagogical work in the context of elementary mathematics classrooms. 
Teaching practice is regarded as a strategic and interested activity since the 
practice of prospective teachers "in schools always works through vested interests, 
both of their own and others' rhetoric of opinions and arguments" (Walshaw, 2004, 



p. 78). In some cases, the institutional practices of the university course and of the 
school involve painful negotiations to produce individual subjectivity: "I wanted to 
introduce new ideas but did not have enough confidence. I just followed my 
associate's plans. I felt I could not try new things as my associate was set in the 
way things were done" (p. 78). Walshaw concluded that the best intentions of each 
prospective teacher can be prevented by "a history of response to local discursive 
classroom codes and wider educational discourse and practices" (p. 78). 

Ensor (2001) analysed the case of Mary, a prospective teacher who expressed 
alignment with teacher educator's ideas, centred on the notion of innovation, but 
who begun to change her opinions during teaching practice in school. Mary started 
to identify herself with some classroom teachers and "to raise questions about the 
applicability of what she had learned in the mathematics method course" (p. 306), 
since some of them argued that their methods achieved results. Working for results 
is a central aim for many teachers and is part of a certain professional culture with 
which prospective teachers are acquainted. 

In contrast to Ensor' s observations, the case of a prospective teacher 
documented by Van Zoest and Bohl (2002) developed in a context of alignment 
between the university programme and the school internship site. The social 
context of the school was supportive of reform curricular materials and teachers 
indicated that they wanted to change their teaching practices. The prospective 
teacher had a profound conviction that the particular curriculum (CPMP) at this 
school contributed to the mathematical understanding of students. According to 
this, when she assumed a new position as teacher in a school with a traditional 
single-subject curriculum, she put a great effort in adapting it "so that students 
would do more of the types of thinking that she believed CPMP demanded" (p. 
281). This shows the importance of communication and negotiation of 
philosophies between teacher education institutes and schools during the 

Looking for Their Professional Development 

One important aspect concerning the development of teachers' professional 
identity is how they assume responsibility for their own professional development. 
There are some studies which start to address this perspective. Olson, Colasanti, 
and Trujillo (2006) described two prospective teachers who accepted leadership 
roles when they began their teaching career. The researchers hypothesised that the 
transformative experiences (cognitive and affective ones) during university 
education promoted their self-efficacy and thus positioned them to assume 
leadership roles. The above mentioned study by Van Zoest and Bohl (2002) also 
described the first year in the career of a teacher who developed an important role 
as a reformer teacher at her new school, and it seems that "the fact that there was a 
social network of jointly-engaged educators working towards the same goals [...] 
had great impact" (p. 284). 

Goos (2005) also presented evidence to illustrate how beginning teachers used 
their (technology-related) expertise "to act as catalysts for technology in schools" 



(p. 56). The beginning teachers in the study showed initiative in trying to develop 
themselves as technology users. Particularly in the case of one of these, the author 
argued that he was "not simply reproducing the practices he observed nor yielding 
to environmental constraints, but instead re-interpreting these social conditions in 
the light of his professional goals and beliefs" (p. 55). Goos emphasized that this 
study shows it is possible that teachers implement innovative approaches from the 
very beginning of their careers. 


The development of a professional identity as (mathematics) teachers is a process 
that is intrinsically connected with their participation in different communities. In 
some studies, attention is given to the activity of the prospective teachers as part of 
one or more communities as they practice teaching in the classroom. For example, 
reflection, as a central process in learning to teach, as presented in the studies 
above, does not exist in a social vacuum. Some of these studies illustrate conflicts 
between the perspectives of different communities involved in initial teacher 
education. For Lerman, "reflective practice takes place in communities of practice 
[...] and learning can be seen as increasing participation in that practice" (2001, p. 
4 1 ), and that involves very strongly the development of a certain professional 


In a longitudinal study, Oliveira (2004) followed four secondary mathematics 
teachers, Cheila, Guilherme, Rita, and Susana, for three years after they finished 
their five year teacher education programme and she tried to characterize the 
development of their professional identity and the role of different contexts and 
processes. Adopting a psycho-sociological model of the person (Gohier, Anadon, 
Bouchard, Charbonneau, & Chevrier, 200 1 ), professional identity is regarded as a 
process that starts from the time the first ideas of becoming a teacher appear. 
Assuming that professional identity develops through a complex and dynamic 
process that faces many constraints and threats with respect to continuity, 
congruence, self-esteem and personal and professional orientation (Oliveira, 2004), 
we choose to illustrate a tension that is pointed in this research as continuity versus 
rupture. The first significant rupture identified in the development of these 
beginning teachers' identities occurred during their teacher education programme. 
All of them recognized that the courses on the didactics of mathematics contributed 
strongly to a change in their perspectives about the teaching and learning of 
mathematics and the mathematics teacher's role. For example, they stressed the 
importance of promoting student-centred teaching methodologies and the use of 
several strategies and resources, in contrast to the teaching style they were used to 
when they were secondary students. Nevertheless, the study revealed different 
levels of rupture and of focus among these four beginning teachers. 



Guilherme expressed a deep change in his perspectives about the nature of 
mathematics as a consequence of the readings and discussions that took place in 
the methods classes. He had been a very successful mathematics student in 
secondary school and at university, but began to see "another type" of mathematics 
and to rethink what it means to teach mathematics. This was a turning point for 
him and he started to see the teaching profession as a stimulating intellectual job, 
one that is constantly changing. 

Rita and Susana changed their own visions about what it means to be a good 
mathematics teacher, one who developed good pedagogical content knowledge that 
focused not only on the students' success but also on achieving more significant 
mathematics learning. They came to see the teaching of mathematics as a much 
more demanding and complex profession than they thought it would be, namely 
that the teacher has to give attention to a lot of aspects beyond teaching the content. 
Cheila also recognized that her vision had been transformed and now considered 
that it is necessary to change old methodologies if students are to be more 
motivated to learn. 

The teaching practicum that the four teachers experienced occupied a full year at 
a school and involved teaching two different mathematics classes. This was a time 
in which the teachers predominantly continued with the ideas they developed 
before, except for Cheila who reached a new turning point. Her expectations that 
the ideas she developed concerning "new methodologies" would have an impact on 
students' motivation remained unfulfilled. This situation caused many doubts about 
the possibility of putting in practice what she learned. It is worth noticing that 
among the four teachers, Cheila was the only one who felt that she did not have the 
support she needed from the mentor to work with her very low achieving students. 

The main changes for these four teachers began when they had to face the first 
year of teaching on their own, in basic schools that did not have any induction 
programme for new teachers. Now they had to regard themselves as autonomous 
teachers. Rita and Guilherme came to understand that they could have an important 
role as teachers and not "merely" as mathematics teachers. Guilherme made every 
effort to know the students well and to attend to their various needs. Rita also 
clearly assumed the role of an educator, who wants to contribute to the social and 
personal development of students. 

Cheila, at this point, expressed a great rupture with the ideas she associated with 
the university programme. She questioned the applicability of the programme's 
ideas in practice and developed a teaching style consistent with the one she 
experienced as a student and in which she succeeded. From there on, her major 
concern was for having professional stability and, consequently, maintaining a 
continuity in her perspectives and practices. 

In Susana's case, there was an initial moment of continuity in the first year; 
however, as time went by she began to question her own perspectives through 
confrontation with other teachers' perspectives and through reflection on her 
incapacity to deal with some difficult situations in the classroom. Since she looked 
at the profession as her natural vocation, this tension created deep doubts about her 
professional and personal projects. 



It is important to stress that most of these beginning teachers had very difficult 
positions in schools that were labelled as "unwanted". However, some of them 
were creative in terms of their identity development, one that was positive and 
congruent from their point of view. 

Besides the teacher education programme, there are many conditions that appear 
to have contributed to the development of these different professional identities. It 
is interesting to note that Rita and Guilherme immediately became involved with 
continuing teacher education and, especially, Rita participated in a research group 
located at the Mathematics Teacher Association (APM, in Portuguese). In contrast, 
Cheila and Susana only participated in some short sessions in schools about 
specific aspects of the teachers' roles. Additionally, Cheila had no reference group 
and Susana was far way from those who constituted her reference group (her 
colleagues from the teaching practice). 

These beginning teachers participated in the same teacher education programme 
but developed very contrasting professional identities (Oliveira identifies four 
different identity configurations). When these beginning teachers talked about the 
perspectives they developed in the programme, superficially they sounded quite 
similar. However, their teaching practices differed markedly. Teacher education 
discourse does not affect all prospective teachers in the same way, as they are 
different people, with diverse expectations, experiences and origins. This study 
also showed that their biographies can reveal much about their beliefs, values and 
knowledge about mathematics and its teaching. 


Learning to be a mathematics teacher is a single trajectory, through multiple 
contexts (Perresini, Borko, Romagno, Knuth, & Willis, 2004), and involving many 
characters. However, in the research we analysed we focused on the learning of 
individual prospective (mathematics) teachers. Programmes that focus on teacher 
education and research on teachers' beliefs show sensitiveness to the individual 
prospective teacher. Usually in this research, "the student teacher is recognized as a 
learner and an active processor of knowledge", one who develops "systems of 
constructs through which they interpret their undergraduate experiences" (Llinares 
& Krainer, 2006, p. 430). However, in the need to theorize about these, research 
sometimes does not attend to the prospective teacher holistically as a person and to 
the social origins of his or her beliefs. It seems that research is now beginning to 
incorporate the fact that beliefs are also contextual i zed (Llinares & Krainer, 2006). 
Studies on prospective teachers' knowledge focus on the individual, but quite 
often these do not show the prospective teachers ' perspectives about what they are 
learning and how, neither do they explain the development of that knowledge, 
taking into account past and present personal experiences of prospective teachers. 
As Ponte and Chapman (in press) notice, it is rare for reflection on self to be 
addressed in research on the development of knowledge of mathematics teaching; 
it occurs mainly as "a by-product". Making tacit knowledge explicit would be an 
important element in their development as future mathematics teachers. In the case 



of prospective secondary mathematics teachers, with a strong mathematical 
background, it is common for them to "value mathematics as important and 
beautiful but lack a critical attitude to mathematics and to teaching itself. 
Elementary teachers often value educational goals but just use "mathematical weak 
conceptions" (Jaworski & Gellert, 2003, p. 843). 

The recognition of the importance of linking theory and practice led many 
programmes to develop frameworks for promoting learning in context. There is 
now a growing body of research aiming to understand how prospective teachers 
"make sense of their beliefs, reflect, and learn while participating in field 
experiences" (Lloyd, 2005, p. 443). 

Research has shown that many prospective teachers are receptive to the reform 
ideas that are presented in teacher education programmes at the universities, but 
interpret these differently. It seems that they develop a professional argot, 
phraseology typical of mathematics education, and some myths about classroom 
reform. When they try to change their role as teachers, they realize that this may 
involve taking on practices that differ the ones they were used to observe. But they 
often do not know how to develop new communication patterns in the classroom or 
to teach for conceptual understanding. Teacher educators clearly have a demanding 
task to help prospective teachers to assume their role in a new classroom setting 
and informed by new theories on learning. 

The process of learning to teach in practice has to do with teachers' perceptions 
of their own knowledge, of what they are able to do or not do, of what in particular 
they think will benefit their students, and so on; and this can be seen as evolving 
their beliefs and their knowledge, as well as themselves as persons. Prospective 
and beginning teachers' dispositions and identities are receiving increasing 
research attention. There is a growing debate about the dispositions for teaching 
and efforts to define and assess them (Borko, Liston, & Whitcomb, 2007). 
Although, if we want to attend to the individuality of the prospective teacher 
regarded as a whole person, it is not fruitful to try to match his or her development 
with a list of dispositions. What some of the studies we analysed suggest is that the 
enormous difficulty involved in learning to teach in the current, challenging times, 
needs to be balanced by a deep consciousness about what it means to be a teacher. 

The paradox for the prospective teacher (and consequently for teachers 
educators) is that teaching is a long term enterprise, meaning that the teacher's 
decisions depend upon other long term decisions and not just of those they can 
make during a limited period of time. Issues of conflict between prospective 
teachers' perspective and those of their mentors raise important ethical questions 
for teacher education, namely, how is it possible to create a good balance between 
promoting the development of a professional identity that is attuned to the 
educational aims of the institution and respecting the professional autonomy of the 
mentors and of the schools were the practicum occurs! 

Research on prospective and beginning mathematics teachers reveals that they 
do not simple reproduce the institutionalized practices in which they teach and that 
it is possible for them to become active agents of their own development (Goos, 
2005); this shows that they are re-interpreting the social conditions of their 



professional context "in the light of [their] own professional goals and beliefs" (p. 
55). We have to recognize that initial teacher education is just the beginning of a 
long journey of professional development for teachers, and that the constitution of 
a professional identity is subject to multiple influences (personal and contextual), 
many of which teacher education can not fully anticipate. Becoming a mathematics 
teacher is not "a sudden move from novice to experienced practitioner on the 
completion of a module or the passing of a test" (Jaworski, 2006, p. 1 89). 

We add some final words about some challenges to teacher education. A focus 
on the individual prospective teacher as learner has a strong parallel with the need 
for the teacher to make an effort to know individually the students in front of him 
or her. But this can be very difficult when teacher educators are responsible for 
dozens of prospective teachers (as also happens with school teachers and their 
students). The advancement of technology has opened new opportunities in teacher 
education. The use of multimedia can facilitate learning from practice. However, 
using these resources gives only a certain picture of mathematics teaching and is no 
substitute for real interaction with students in the classroom. The use of the Internet 
to promote virtual interaction is another possibility for teacher education but there 
is still much to understand about how people "perform" in these scenarios. At the 
same time, we observe in educational research an increasing interest in looking at 
learning as participating in communities of practice, and it becomes clear that it is 
not possible to study the individual prospective teacher's learning without 
considering the contexts where it takes place. However, in this chapter, we 
intended to illustrate how learning to teach is also an idiosyncratic process and to 
review research providing evidence for this. As the studies analysed in this chapter 
show, much of the research on initial teacher educator is done by the teacher 
educators themselves (see also Adler et al., 2005). Therefore, there is a strong 
perception that the knowledge accumulated can be used readily in the design and 
development of new programmes and courses and (most importantly) in the 
interactions between teacher educators and prospective teachers. 


We wish to gratefully acknowledge the helpful comments and feedback on former 
versions of this chapter provided by Gilah Leder, Heinz Steinbring and Terry 
Wood, and the continued encouragement and feedback from Konrad Kramer. 


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Helia Oliveira 

Centra de /nvestigacao em Educaqao da FCUL 

University of Lisbon 


Markku S. Hannula 

Department of Applied Sciences of Education 

University of Helsinki 





Studies on Their Professional Growth 

This chapter focuses on the professional growth of practising teachers. In the first 
part, two examples illustrating major trends of current research allow to launch 
the discussion. We then present our methodology and the main results from our 
review of literature. Finally, we conclude on three main issues and perspectives: 1) 
the change of paradigm proposed in practising teacher education poses new 
teaching problems, thus it seems important to study teaching in its context; 2) when 
teachers have to change because of an external constraint, some changes may 
occur, but they are not always the "wanted" ones; we need better understanding of 
deep components of practice; perhaps the study of the stabilization of practice 
during the first years of career may help progress; 3) it seems important to 
construct concepts or systems capable of taking into consideration the variety of 
teachers ' work (from planning to classroom interactions). 


This chapter focuses on the professional growth of practising teachers. We are 
interested in the evolution of teachers' practice, knowledge and beliefs, as well as 
the constraints, dilemmas and difficulties they face in adapting their practice to 
students or to a new curriculum. Though the constructs and theoretical frameworks 
are often different, we distinguish three main issues: 1) How are teachers' 
conceptual, belief and knowledge systems organized in relation to their practice? 2) 
How are these systems to be changed in order to improve teaching? 3) How do 
teachers' practices change "naturally" and in what ways do teachers learn from 
their own practice? The first two of these questions are close to issues raised by 
Adler, Ball, Krainer, Lin, and Novotna (2005) in their survey on mathematics 
teacher education, whereas the third issue, at the core of our chapter, is scarcely 
addressed in the literature. 

Part one of this chapter presents our general point of view regarding the 
problem. Two examples from our own recent research are included as a point of 
departure. We then specify the methodology we used in conducting our review of 
the literature. In the third and fourth parts, we state our research questions and 
present our results. We then move on to provide a more in-depth discussion of 
those issues with more recent research about actual practices and their 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, $5-59. 
© 2008 Sense Publishers. All rights reserved. 


development. Finally, we draw some provisional conclusions as to what new 
questions, concepts and methodologies might be considered. 


Our Point of View 

One's learning as a mathematics teacher is a lifelong learning process which starts 
with one's own experiences as a learner of mathematics, later supported by both 
prospective and practising teacher training. In this chapter, we view professional 
growth as a progressive transformation of mathematics teachers' actual practice in 
relationship to their individual and professional experience, their knowledge and 
their beliefs or conceptions about mathematics and mathematics teaching. In the 
literature, terms such as teacher development, changes in teachers' practice or 
teachers' learning are often used indiscriminately, though they are not equivalent. 
For practising teachers, teacher development can be seen through the changes that 
occur in their practice. Nonetheless, recent research seems to indicate that teachers 
may learn about teaching, for example about student's logical development or 
challenging problems but not change their practices, or they may change their 
practices without really new thinking. 

Behind the three research issues considered above, important questions arise: 

1) Is it possible to study only beliefs, or knowledge, or practice? What 
relationships between these concepts are most relevant? For example, beliefs can 
influence practice, but practice can also influence beliefs. How can we study 
systems of beliefs and knowledge and their relationship to practice? Research 
issues depend on these implicit or explicit assumptions. Many models have been 
constructed to explain interactions between different views of teachers. In 
reviewing the literature, we see that during the last decade a shift has occurred 
from research describing one issue (beliefs, or knowledge or practice) to research 
taking greater account of complexity and the relationship of these constructs to 

2) In what sense can we speak of professional growth? What constitutes good 
teaching? Very little research goes so far as to study the effect of teaching on 
students' learning. It looks like as if an implicit agreement exists about what is 
good teaching (see e.g., Wilson, Cooney, & Stinson, 2005) and such teaching 
produce students' best learning. In many studies, particularly in US, the implicit 
reference is to the NCTM standards without questioning its effect on students' 
learning, particularly in the case of socio-cultural differences between students. 
Another possible reference for professional development among research is the 
participation in a community of practice (and development of this participation). 
Otherwise, the reference may only be the researcher's conception of good teaching. 

3) What is meant by "natural" change? The difficulty in accessing natural 
teacher growth without modifying it from outside may explain why there are so 
few studies on this topic (e.g., self-study, often from teacher educators or teachers 
involved in research projects). Teachers may decide themselves to register for a 



practising teacher education course. This decision is an indication that they wish to 
improve their practice but the changes are influenced by the teacher education 
program and are not "natural". 

4) The relevance of making a synthesis of questions and results achieved from 
very different theoretical perspectives and with different aims may be questioned. 
Theory may be a theorisation of practice: assumptions on learning and on effective 
teaching; in such case, research consists in elaborating and testing such a theory of 
practice. In other research, theory is a construct in order to analyse, understand and 
explain practice and relationships between teaching and learning, for any style of 
teaching. Sometimes, both positions co-exist in the same research, in particular in 
the case of collaborative research (e.g., Scherer & Steinbring, 2006). In this 
chapter, we consider mainly research the aim of which is to study professional 
practice in order to understand it and thus what may make it evolve. 

Two Examples to Launch Our Discussion 

Example 1: Enhancing our understanding of stability of practice. We refer here to 
a research which was done in 2004-2006 (Paries, Robert, & Rogalski, in press). 
This work aims at identifying specific difficulties that experienced mathematics 
teachers' face when changing their own practices. In the following, we discuss one 
of the two cases presented in the article mentioned above. 

This case investigates two lessons given by the same teacher at grades eight and 
nine in "ordinary" classes 1 . The lessons focused on the same kind of geometrical 
activities for both groups of students (the resolution of problems just after a lesson 
on a new theorem). To exhibit stability of practice both mathematical tasks and 
lesson management were analysed. 

There was a real difference in how the tasks were proposed in these classes. In 
the first class, students had to conceive of some steps on their own before applying 
corresponding theorems (once the Pythagoras theorem, once the converse, and then 
a theorem on the sum of angles). In the second class, the problem applying 
Thales's theorem 2 was more direct. The difficulty, however, was to write "x" 
instead of a formal length "EM" in an equality. There were few differences in 
terms of classroom management. The activities explicitly organised by the teacher 
for her students and the order in which the activities are carried out are the same in 
both groups (drawing the figure, looking for a strategy and then for the resolution 
and correction). We note the similarity in the length of each activity (including the 
total), even if there are slight differences. At that point, it is important to assess the 
previous comment: this stability of classroom management has been confirmed by 
the teacher as typical for this kind of problem when she was asked about the 
representativeness of these videos. So we can assume that they represent the usual 
way of working for this teacher on analogous problems. 

1 Students work in class, at home and there are no special problems of discipline. 

2 The theorem of proportional lengths made by a parallel to one side of a triangle. 



Table I. Lesson plan (with time in minutes) 

On Pythagoras' 

grade 8 

On Thales' 

grade 9 

Work organised 
by teacher for 

First question 

(1) beginning, (2) 


Second question 

Total for 
the grade 8 

Draw the 





More than 

Looking for a 
strategy to 
solve the 

Individual then 



Collective (I) 



Looking for the 


In two steps 


4' 10 

Correcting and 
recopying the 

The teacher 

writes on the 

blackboard at 




Two students chosen 
by the teacher write 
successively on the 

One student 
writes on the 





The relationship between tasks and the students' activity are different in the two 
lessons, not only in terms of the tasks themselves but also in terms of the teacher's 
management 3 . As the first problem, which can be more difficult than the second 
problem, required more steps, it was impossible for many grade 8 students to come 
up with something during the 8 minutes they had to look for a strategy. The teacher 
relied on some isolated students' proposals and finally explained the whole strategy 
to the students who, for the most part, did not find anything. She then wrote the 
three steps on the blackboard. The students then had to work on three easier tasks. 
Perhaps, if the teacher had allowed more time or if the students had had the 
possibility of working in small groups, more students would have begun to move in 
the direction of a resolution. 

One can argue that the teacher does not mind changing the task in that way. It is 
not clear because the teacher's last sentence in grade 8 is "You see [...] you are able 
to solve a problem without intermediate questions." So, even if the teacher does not 
mind this reduction of students' activities, it does not change the fact that her 
(stable) management style is incompatible with letting students work by 
themselves on such complex activities. On the contrary, in grade 9, more students 
find the strategy. The difficulty of calculating with x is widely anticipated by the 
teacher and it seems that a lot of students get involved in it. Maybe the teacher's 

5 We use here the term "management" to precise the part of practice by which the teacher manages the 
students' mathematical activity. 



management is well adapted to such a problem because this exercise was an old 
one for the teacher and the other problem was more recent, selected for the purpose 
of providing complex activities. What can be deduced from these analyses? 

The stability of experienced mathematics teachers' practice concerns, first of all, 
the management of the session at the scale of lessons. In other words, tasks are 
easier to change than management (see e.g., Cogan & Schmidt, 1999). Even if 
tasks adopted by teachers for students change, due to, for example, new curricula 
or new standards, this does not ensure that the expected consequences for students' 
work occur, because of no variation in the way they choose to manage their 
classroom. In the case where students have to work in a precise way on some (new) 
tasks to benefit from new activities, as finding steps by themselves before applying 
a theorem, we can guess that nothing will occur in the class if the teacher's usual 
management does not let students work in a compatible way. Consequently, it may 
explain why it is so difficult for teachers to change their own practice. Maybe, the 
management of their practice is so stable that in order to change it has to be 
revealed by somebody else, for example, by mutual observation, and explicitly 
discussed to be questioned (see also Nickerson, this volume). 

Example 2: Conditions leading to the transformations of the interpretation and the 
intervention. Previous research of DeBlois and Squalli (2001) leads to the study of 
how discussions among teachers may help them transform their understanding of 
their students and develop their practice. 

A first study (DeBlois, 2006) was done with elementary school teachers 
concerning the errors contained in students' written products. It showed, for 
example, that when teachers compare the task to all the tasks usually proposed, 
they evaluate the influence of the student's work habits. This type of reflection 
makes the error "logical". It is no longer synonymous with a lack of attention on 
the part of the student. The error becomes an extension of the procedures known by 
students. This reflection then leads the teacher to choose a form of intervention that 
allows students to break with their usual work habits. This study was continued 
with teachers at the early high school level. Seminars allowed teachers to discuss 
the errors contained in students' work. At the beginning, teachers expressed 
themselves in an affirmative way when they described the students' production or 
when they talked about the teaching offered and other parts of environment to 
which they are sensitive. During the discussion, they were entering a process of 
dissociation from the teaching offered. This dissociation allowed them to review 
the tasks performed with and by the students in the classroom. 

For example, four teachers (D, K, M, J) discussed a 13-year-old student's 
production (Figure 1). At first, one teacher could not find any justification to 
explain the strip diagram. Another teacher explained that/how he had worked on 
the vertical and horizontal diagrams with the students. He added that he informed 
the students about what could be asked. The problem exposed in the student's 
production became afterwards proof of confusion between the two elements of 



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Figure I. Student's production. 

Time and a collective decision became the means of the student's understanding 
as shown in this excerpt: 

D - 1 can impose you, [precise] what I want. I can say: trace the one that is 

appropriate and you will have to choose the one that is appropriate. 

K - If you had shown the two of them, they could have mixed them up [...]. 

D - Because different situations required two different diagrams. 

D - Give a class showing the possibility of making a horizontal or vertical 

diagram and take a decision in group. In two groups, the choice of the strip's 

form was different, one chose the vertical diagram and the other chose the 


Other teachers considered the error as a way for students to regulate how to get 
good answers that, by itself, is part of the student's learning process. 

M - Maybe they wanted to do a horizontal strip first? The zero should have 

been there. 

M - It is possible. He could have changed his mind at the last minute. 

The fact that a teacher brings out the notion of regulation weakens for the group 
the hypothesis of an automatism suggested at the beginning of the discussion. The 



research method undoubtedly influenced the discussion. In fact, asking teachers to 
describe their students' written products was above all an invitation that freed them 
from concerns they had. Indeed, the teachers' interpretations seemed to change as 
other possible clues concerning the origin of the error were explored. Thus, in this 
case, among the possible interventions, we find the wish of knowing the student's 
representation of a graphic and a diagram. This way, we can consider the 
discussion between colleagues as an environment for re-examining students' 

In the excerpt studied, it was the moments of intense interaction that modified 
the teachers' interpretation and allowed the understanding of the student's 
production, resulting in an effect on the planned interventions. Their sensitivity 
appears mostly in regard to the particular learning conditions, to students' 
uncommon procedures and to institutional knowledge. The teachers agree that, 
despite the confusion that led to an inversion of the representation, the final result 
makes sense. The analysis shows that, from time to time, the researcher, while also 
playing her role as a mathematics teacher educator, uses teachers' affirmations as a 
starting point to call attention to other hypotheses or observable facts in the 
student's production which were ignored until then. This way, the interaction 
produced between teachers and the researcher played an important role in the 
process of transformation. 

These two examples illustrate two major trends of current research: On the one 
hand, there are diagnoses of difficulties for transforming practice, they involve 
beliefs, classroom management, or cultural and institutional constraints. On the 
other hand, research presents some ways of overcoming these difficulties, 
including new concepts for analysing teachers' appropriation of teacher education. 


In order to define the three areas of research mentioned at the beginning of this 
chapter, we started a literature review with the following interrelated questions in 

1. What is studied (practice, knowledge, beliefs, relations among them, how to 
change them, how they can change)? We also tried as much as possible to specify 
the cultural and political context as main factors of practice. 

2. What factors are taken into account for professional growth and their effects 
(institutional or cultural factors: reforms, new technologies; later effects of 
previous training; research participation; individual factors: learning from 

3. What theoretical framework, implicit or explicit hypotheses, and what 
methodologies are used to gain access to individual professional growth? 

4. What are the findings? We will distinguish the results concerning primary and 
secondary school teachers. Indeed, their conditions are not the same, neither 
concerning mathematics (their learning of mathematics as students which may 



influence their practice as teachers) nor with regard to teaching (the former teach 
many subjects and not only mathematics). 

The key words used in our search of literature were: teachers' experience, 
teachers' beliefs, teachers' practice, teachers' learning, professional knowledge, 
and practices of mathematics teachers. We used also handbooks and synthesis 
articles as well as their bibliographies in selecting research concerning the 
professional development of practising teachers. Moreover, a variety of reviews 
were systematically studied. For the selected papers, we drew up a summary taking 
into account issues I to 4 above. This allowed us to define categories used to 
present the third and fourth parts. 


A Recent Emergence 

The interest of scholars for teachers as object of study is recent. Fifteen years ago, 
Hoyles (1992) deplored the scarcity of teacher- focused research and appealed to 
develop it. Her request was granted: For example, Sfard (2005), Ponte and 
Chapman (2006) noticed a growing interest in this subject and a surprising growth 
of papers on teachers' practice since 1995 (see also Krainer, volume 4). Similarly, 
Margolinas and Perrin-Glorian (1997) noticed that research concerning teachers' 
action has developed in France since 1990 and a book on Teacher Education 
emerged from the first European meeting on Mathematics Education (Krainer, 
Goffree, & Berger, 1999). 

Several reasons may explain this growing interest in teachers' practices and the 
way they can evolve: results that did not answer the teachers' questions but the 
researchers' questions; research did not consider enough constraints of the class. 
Thus, mathematics education researchers felt the need to better understand 
teachers' conceptions of mathematics teaching and their various constraints to 
conduct their teaching. These new questions required an extension of the 
theoretical frameworks. For example, in France where the question of the teacher 
was posed early, the theory of didactic situations in mathematics (Brousseau, 
1997), particularly the notion of milieu, was refined to study "ordinary" classes. 
The teacher's action on 4 the milieu of a didactic situation is a key issue to 
understand his or her role in class (Bloch, 2002; Hersant & Perrin-Glorian, 2005; 
Margolinas, 2002; Margolinas, Coulange, & Bessot, 2005; Perrin-Glorian, 1999; 
Salin, 2002). Extending the notion of didactic transposition, a new framework 
emerged in order to help describe the organisation of study at school (Barbe, 


In the theory of didactical situations (Brousseau, 1997), the milieu of a didactic situation is the part of 
the context that can bring a feedback to student's actions to solve a problem. The teacher can act on the 
milieu bringing some new information or new equipment, for example, asking a question or giving a 
compass; acting on the milieu, he changes the knowledge needed to solve the problem. One can refer to 
Warfield (2006) for an introduction to this theoretical framework 



Bosch, Espinoza, & Gascon, 2005; Bosch & Gascon, 2002; Chevallard, 1999; 
Chevallard, 2002). Other scholars endeavoured to model teachers' didactic action 
using a combination of these two theories (Sensevy, Mercier, & Schubauer-Leoni, 
2000; Sensevy, Schubauer-Leoni, Mercier, Ligozat, & Perrot, 2005). The didactical 
and ergonomical twofold approach of Robert and Rogalski (2002, 2005) copes in 
another way with the complexity of practice. 

In Quebec, different program reforms have led researchers to develop new 
contexts for practising teacher education. The theoretical frameworks of SchOn 
(1983), Lave (1991), Erickson (1991), and Bauersfeld (1994), contributed to the 
development, experimentation and analysis of collaborative research (Bednarz, 
2000). In this perspective, the teachers' practice is the starting point of the training. 
The didactic content is more a tool for training than a corpus of knowledge to be 
transmitted. In this way, teachers could develop a deep understanding of 
mathematical knowledge. Recently, Bednarz (2007) observed that research was 
suffering from an absence of a teacher-oriented database which allows for a 
thorough consideration of classroom situations. 

At the same time, at the international level, sociocultural theories (D'Ambrosio, 
1999) and references to Vygotsky's work were spreading as well as a new interest 
for teacher and students' interactions. Several models appeared; theoretical efforts 
were done to control the teacher's role or action inside theory in order to 
understand teacher's work. However, it is not sufficient to understand how to 
change practice; for that, it is necessary to understand teacher's work in and of 
itself. Thus, research concerning teacher change is even more recent, involving 
various questions and methods referring to psychological or sociological 
perspectives (Richardson & Placier, 2001). The recent 15th ICMI Study 
Conference on The Professional Education and Development of Teachers of 
Mathematics (2005) was a crucial moment to discuss this theme. 

Understand Teachers ' Practice and Teachers ' Growth 

Previous experience of learning and teaching strongly influences current teaching. 
Though we cannot separate beliefs, knowledge and practice to understand teachers' 
practice or teachers' growth, the aim of this section is to clarify these notions and 
their use to better understand the way practising teachers may evolve. 

Beliefs. Research on beliefs was first carried out from psychological perspectives 
and beliefs were treated as cognitive phenomena (see e.g., Thompson, 1992). Some 
studies used questionnaires, others focused on describing teachers' beliefs or 
conceptions in relation to a particular aspect of teaching or learning mathematics, 
problem solving, students' errors or technology for many of those involving 
practising teachers. The results of these studies led researchers to appreciate the 
complexity of the notion of beliefs (Ernest, 1989; Jaworski, 1994; Mura, 1995). 

More recently, their contextual ised nature and their social origin were 
considered (see e.g., Gates, 2006; Leder, Pehkonen, & TOrner, 2003; Llinares & 



Krainer, 2006). Nevertheless, some scholars, for example in France (Robert & 
Robinet, 1996) referred earlier to social representations. Be that as it may, beliefs 
and attitudes about mathematics, mathematics teaching and the role of the teacher 
were regarded as a main factor influencing teachers' teaching and their learning 
processes about mathematics teaching (see also Oliveira and Hannula, this 

Knowledge. Many studies have attempted to know more about teachers' knowledge 
for teaching mathematics, and its effect on students' learning. 

Early research showed that there was no clear relationship between the number 
and the level of mathematics courses completed by teachers and students' 
achievement. Then research focused on a specific mathematical content, identified 
deficiencies and misconceptions in teachers' mathematical content knowledge on 
many topics, generally for prospective teachers or primary school teachers. A 
number of papers point out explicitly implications for teacher education but the 
relationship with teaching was not evidenced. Thus researchers grappled with the 
question of what would constitute conceptual understanding for teachers and felt 
the need to extend the theorisation of teachers' mathematics knowledge by 
including teachers' knowledge of mathematics for teaching (Ball, Lubienski, & 
Mewborn, 2001 ; Ponte & Chapman, 2006). 

Shulman (1986) was the first to call attention to a special kind of teacher 
knowledge that linked content and pedagogy: pedagogical content knowledge 
(PCK). Research into PCK identified the necessity of teachers' awareness on 
students' difficulties in specific subjects. Later, the notion of PCK was often 
combined with other theoretical constructs. Studies involving PCK showed an 
effort in establishing a critical perspective. For example, Ma's study (1999) 
described what she called "profound understanding of fundamental mathematics" 
to explain the difference between American and Chinese teachers. That said, Ball, 
Lubienski, and Mewborn (2001) claim that the descriptions of teachers' knowledge 
do not necessarily illuminate the knowledge that is critical to good practice. 
Moreover, distance remains between studies on teachers' knowledge and on 
teaching itself. Ball, Bass, Sleep, and Thames (2005) developed an extension of the 
notion of PCK. They identified four domains of which two are close to the 
Shulman's work: knowledge of students and content, knowledge of teaching and 
content, and two are new: common content knowledge and specialized content 

For Suurtamm (2004), professional development necessarily passes through a 
deepening of one's mathematical knowledge in order to answer the students' 
questions and to guide them in their exploration process. Many authors confirm 
this vision, by linking risk taking, self confidence, teamwork and the use of 
appropriate resources with teachers' professional development (Brunner et al., 
2006; Carpenter & Fennema, 1989). Even (2003) recalls that insofar as learning is 
a personal construction, the construction of mathematical knowledge does not 
necessarily reflect instruction. Some common experiences (subjective and 
sociocultural) will favour a common signification contributing to deepen reflection. 



Thus, the research has revealed the complexity of mathematical knowledge for 
teaching, its links with mathematical knowledge thus indicating "a shift away from 
regarding mathematical knowledge independent of context to regarding teachers' 
mathematical knowledge situated in the practice of teaching" (Llinares & Krainer, 
2006, p. 432). 

The research approach developed mainly in France around the basic notions of 
the Theory of Didactic Situations does not focus on actors but on teaching and 
relationships to mathematical knowledge. Rather than pedagogical content 
knowledge or knowledge on mathematics teaching, Margolinas et al. (2005) use 
the notion of "teacher's didactic knowledge". This notion is defined as part of 
teacher's knowledge "which is related to the mathematical knowledge to be 
taught". They prefer this notion because, as Steinbring (1998) has also stressed, 
mathematical knowledge and pedagogical knowledge cannot be separated for 
teaching. Indeed, crucial questions for teacher education now are 1) what special 
form of mathematical knowledge is fundamental for teaching? and 2) how can 
teachers not only acquire such knowledge in order to use it effectively in the 
classroom but go on developing it over the course of their teaching careers? We 
will see in part 4 how recent research addresses this question. 

Practices. There is a large variety of studies on teachers' practices referring to 
various theoretical frameworks and methods. Some of them describe practice in 
terms of indicators such as amount of time spent on lesson development, types of 
problems selected during development, teacher's types of questioning and so on. 

Other studies focus on the relationship between the structure of the lesson and 
the teachers' understanding of a specific mathematical content and others are 
biographical studies. Among them, some studies are linked to education reform 
efforts or to the quest for effective practices. There are also many studies at a 
microdidactic level which study classroom interactions, with an interest in the 
language used, the nature of classroom discourse, the role that the teacher plays in 
classroom discussions, the identification and characterisation of interaction 
patterns. We can notice that teachers' practices and research on teachers' practices 
strongly depend on the cultural and political context (e.g., Cogan & Schmidt, 
1999). We also must consider results coming from international evaluations like 
TIMSS and PISA. These evaluations may have some influence on practices. For 
example, some countries may develop students' training in problem situations like 
PISA to have best performance. Thus, the teachers' practice, the curriculum and 
the studies could nowadays be influenced by international evaluation (Bodin, 2006; 
DeBlois, Freiman, & Rousseau, 2007). 

Ponte and Chapman (2006) point out that the notion of practice used in research 
has evolved. Mostly regarded as actions or behaviours in early studies, practice 
includes later what the teacher does, knows, believes and intends. Boaler (2003) 
and Saxe (1999) consider the notion of stability and recurrence of practices. Saxe 
emphasizes the socially organised nature of these practices; Boaler considers not 
only activities but also norms. We consider with Ponte and Chapman (2006, p. 
483) that "teachers' practices can be viewed as the activities they regularly 



conduct, taking in consideration their working context and their meanings and 

Relations between Beliefs and Practices 

Initially, it seemed that one way to improve practice was to improve beliefs and 
knowledge. But research has shown that a change in beliefs does not necessarily 
entail a change in practices especially for practising teachers. Many studies 
focused, directly or indirectly, on the relationship between beliefs or conceptions 
and practices. These studies found inconsistencies between them, particularly when 
teachers were faced with innovation, notably involving computers. Nevertheless, 
these inconsistencies may be apparent only. For example, Skott (2004) explains 
these inconsistencies in terms of the existence of the multiple motives of teachers' 
activity, experienced as incompatible. These inconsistencies may thus be seen as 
situations in which the teachers' priorities are dominated by other motives, maybe 
not immediately related to school mathematics, for example, developing students' 
self-confidence. Vincent (2001) and DeBlois and Squalli (2001) talked about 
preoccupations about the space and the time of learning. Moreover, Lerman (2001) 
criticises past research on beliefs and comparison with practice, including his own, 
because it does not take sufficiently contexts into account: for example, the 
interview context is different from class context. He suggests that "whilst there is a 
family resemblance between concepts, beliefs, and actions in one context and those 
in another, they are qualitatively different by virtue of these contexts" (Lerman, 
2001, p. 36), and that "contexts in which research on teachers' beliefs and practices 
is carried out should be seen as a whole". More recently, Herbel-Eisenmann, 
Lubienski, and Id-Deen (2006) distinguish local and global changes, taking into 
account the importance of curricular context for local changes. Thus, students' and 
parents' expectations and desires as well as the curriculum materials may influence 
the teacher adopt local adaptations not really compatible with his global beliefs. 


What Actually Changes, What Resists? What Means Seem Effective? 

Two main questions on practices strongly concern teacher education: (I) what 
constitutes good practice for students' effective learning? and (2) how can 
teachers' practice be improved? For the first question, research often gives an 
implicit answer supported by constructivist or socio-constructivist theories 
supporting practice that let a large place to students' action. For the second 
question, the research to better understand teachers' practice concludes on the 
complexity of teachers' practice so that it is now widely recognized that 
professional development programs that attempt to achieve real changes in 
classroom practices must address teachers' knowledge, beliefs and practice. 
However, the nature and genuineness of changes further complicates this question. 
The question of efficiency is very difficult to evaluate; actual and deep changes 



require time. Moreover, to find efficiency in students' achievement, one has to be 
sure that the planned changes in practices were achieved (Bobis, 2004; Sullivan, 
Mousley, & Zevenbergen, 2004). Other reasons could explain these difficulties: the 
methodology used, the paradigm adopted, the cultural factors and the context, and 
teachers' engagement. 

Thus, an important methodological problem occurs to know what changes and 
what is resistant to change. How can we compare or measure changes that occur in 
teachers' beliefs, knowledge or practices? Teachers' answers to questionnaires are 
not easily interpretable into actual changes made. Class observation, more often 
used in recent research, is very time-consuming such that only a few cases can be 
studied. Some researchers adopt a mixed approach, asking teachers to react to class 
contexts, for example using video clips of classroom excerpts. 

A common feature of reforms in mathematics education carried out in the last 
decades is the change in relative emphasis from mathematical products to 
processes, with a greater importance given to individual processes. The student is 
expected not only to learn predetermined concepts and procedures but also to 
become involved in genuinely creative individual and collective processes of 
investigating, experimenting, generalising, naming and formalising. This 
adaptation supposes a rather different and more difficult role for the teacher. The 
teacher has to adopt a certain interpretative stance; to engage in reflexive activity 
enabling him to a flexible use of interactions with the students, what Skott (2004) 
named "forced autonomy". He observed some novice teachers claiming teaching 
priorities inspired by the reform. Their classroom practices were in line with these 
priorities most of time but not at other moments, identified as critical incidents, 
where teachers are "playing a very different game than one of teaching 
mathematics". In this type of research, some recent work tries to specify new tasks 
for students or/and new tasks for teachers, to get students to become more engaged 
and more effective problem solvers (Doerr, 2006). Recent research also tries to 
describe the diversity of ways of adoption of this new "job", sometimes in relation 
with an appropriate (or hoped so) teachers' training program (Herbst, 2003). Even 
though teacher education programs have gained some positive results, much 
research highlights the difficulties for teachers to change their practices in a deep 

Other recent work tries to understand the difficulty to change mathematics 
teachers' practice, and how their beliefs are implicated in a complex way in 
practices, at different levels, combined with social or cultural considerations. For 
example, Arbaugh, Lannin, Jones, and Park-Rogers (2006) studied 26 secondary 
teachers using a problems-based mathematics textbook Core-Plus. They conclude 
that adopting a problem-based textbook series and using it in a classroom is not 
enough, in itself, to have an effect on teacher instructional practices - to get them 
to teach in a more reform-oriented manner, according to the strength of previous 
habits and beliefs. Wilson, Cooney, and Stinson (2005) reveal some subtle aspects 
of these beliefs in what constitutes good mathematics teaching and how it develops 
for teachers: they find considerable overlap between the teachers' espoused beliefs 
and the writing in NCTM standards documents as well as important differences, 



linked to these teacher-centred/student-centred classroom conceptions. Moreover, 
the teacher's view and the researcher's view of change may be different. For 
example, Sztajn (2003) shows the way in which elementary school teachers adapt 
their practices to their students' needs, that is to what they think their students' 
needs are, and at the same time being certain that they are acting according to 
current reform visions adapted as much as possible to their specific students. 

Thus, professional development efforts often result only in surface changes. We 
can notice with Tirosh and Graeber (2003) the importance of cultural factors and 
the environment of teachers. Some research reports success of individual teachers 
changing their practices though their colleagues do not (e.g., Koch, 1997). But 
successful change is more likely to occur when simultaneous attention is given to 
changing the system in which teachers work. Many studies have noted the value 
derived from discussions with colleagues who are experiencing similar concerns 
and can provide ideas for solving problems encountered in change (e.g., Kramer, 
2001). Students themselves may resist (Brodie, 2000; Ponte, Matos, Guimaraes, 
Leal, & Canavarro, 1994). Thus it may be easier to change practices with a 
program involving all the teachers in a given school (e.g., Sztajn, Alex-Saht, 
White, & Hackenberg 2004); nevertheless, volunteer teachers will change more 
easily; research projects often balance between these two options. 

Indeed, student achievement is the main goal of the vast majority of efforts to 
change classroom practice and the main motive for teachers to try to do it, 
especially at a large scale (see e.g., large programs for primary schools in 
Australia: Bobis, 2004; Sullivan et al., 2004) or with low-achieving students in 
South Africa (Graven, 2004). 

Taking into account the fact that deep changes take a long time, Franke, 
Carpenter, Levi, and Fennema (2001) also addressed the question why some 
teachers continue to develop their practices when teacher education programs are 
over. Their study, a follow-up of the Cognitive Guided Instruction (CGI) program 
for practising teachers (Carpenter & Fennema, 1989), shows that some teachers of 
this program were engaged in a generative growth. They suggest that focusing on 
students' thinking is a means for engaging teachers in continued learning and that 
helping teachers' collaboration with their colleagues can support it. Graven (2004), 
in a study in the Lave and Wenger's theoretical framework of community of 
practice, adds the notion of confidence as both a product and a process of learning, 
in relation to teacher learning as "learning as mastery". It seems that the notion of 
confidence could help particularly in this case, involving teachers with a low 
(sometimes even absent) mathematical background in their teaching preparation 
and students coming from low socio-cultural backgrounds. However, engaging 
teachers in learning to examine mathematical tasks using the Level of Cognitive 
Demand criteria supports both a growth in pedagogical content knowledge (ways 
of thinking about mathematical tasks) and a change in practice (choosing 
mathematical tasks) proved to be a non-threatening way to start teachers thinking 
more deeply about their practices (Arbaugh & Brown, 2005). 

Nevertheless, an important condition is the teacher' involvement in the program 
during its unfolding and after it. Doerr and English (2006) show that the modelling 



tasks were actually a means used to bring about change in teachers' practices, even 
if that change is not the same for any two teachers. Wood (2001, p. 432) attempted 
to find out if teachers were actually learning by investigating their process of 
reflection, namely "how teachers use reflective thinking in their pedagogical 
reasoning and how their thinking relates to changes in teaching". She confirms that 
teachers develop differently while giving some insight into how this difference 
continues. But the challenge remains to explain why this development is different 
among teachers or to know if another choice in approach to teacher education 
would produce another effect. The study of Empson and Junk (2004) is devoted to 
an assessment of a training program for elementary teachers, novice and 
experienced teachers alike, based on knowledge of students' mathematics. It shows 
that teachers' knowledge of students' non-standard strategies is broadest and 
sometimes deepest in the chosen topic, even if it depends on the teachers; the 
teachers use information given in their training, some of them even extended their 

To change practices (in a broad sense), the importance of reflection on practice 
as well as collaboration and discussion among colleagues about concrete cases (for 
example excerpts of videos, particularly critical classrooms incidents) is nowadays 
stressed (e.g., Nickerson, this volume). Nevertheless, as Llinares and Krainer 
(2006, p. 444) comment: 

At the present time, we need to understand better the relationship between 
these instruments [reflection and collaboration] and teachers' different levels 
of development, as well as the changes in teachers' practice. 

Spontaneous Changes. How Do Teachers Learn through Teaching? 

Interest of scholars in the individual and spontaneous changes in teachers' practice 
is quite recent. Moreover, questions and methods used are different: Searching why 
and how teachers change without questioning if it is in a desirable direction 
(change is not always desirable, especially at the end of their career), studies have 
found that teachers are always changing. They refer mainly to psychological 
background and methods are often based on case studies looking for relationship 
between biography, professional experience and professional knowledge 
development. Results show mainly stability of style with certain flexibility. Large 
differences between teachers are observed, some of them being more able than 
others to frame puzzles stemming from practice (Richardson & Placier, 2001). For 
example, Sztajn (2003) analyses the case of Helen, an elementary school teacher 
with 31 years of teaching experience, who thinks that her mathematics teaching has 
greatly improved during her career. Nevertheless, the same fact may be seen as 
change for the teacher and as stability by the researcher if actually the students' 
activity is not really changed. Thus, Sztajn stresses the difference of scale between 
teachers' and researchers' views about change and the necessity for researchers to 
understand teachers from the teachers' point of view. 



Most scholars share the view that teachers increase their understanding of 
learning and teaching mathematics indirectly from their practice, through years of 
participating in classroom life (Stigler & Hiebert, 1999). However, even though 
"direct and indirect learning are interrelated and depend on each other" (Zaslavsky, 
Chapman, & Leikin, 2003), little research concerns indirect practising teacher 
learning (embedded in practice). Such research may consist in self-analysis carried 
out by teacher educators of their own practice as teachers (e.g., Tzur, 2002) or joint 
analysis of a teacher and a researcher (e.g., Rota & Leikin, 2002) or analysis by a 
researcher of teacher's learning from class observation (Margolinas et al., 2005). 
Tzur (2002) conducted, as researcher-teacher, a teaching experiment in a third- 
grade classroom observing his own improvement of practice. His reflection is 
similar to Ma's (1999) and raises the question if and how a teacher (not researcher) 
can lead this kind of reflection in his class. Rota and Leikin (2002) studied the 
development of one elementary school beginning teacher's proficiency in 
managing a whole class mathematics discussion in an inquiry-based learning 
environment. They found a large growth inflexibility for the teacher, much more 
attentive to the students, without any professional development intervention, except 
the existence of their research. They also stress the difficulty of teaching a teacher 
when to apply a particular teaching action. 

More research is needed on teachers' learning through teaching, especially for 
beginning teachers. However, some models explain how such learning may occur, 
and enhance in some way previous results, as they suggest that a teacher cannot 
"see" what he is not prepared to see. This explains that a deep professional growth 
cannot occur in some cases without some external interventions. For example, 
Zaslavsky et al. (2003, p. 880) refer to Steinbring's model (1998): 

According to this model, the teacher offers a learning environment for his or 
her students in which the students operate and construct knowledge of school 
mathematics in a rather autonomous way. This occurs by subjective 
interpretations of the tasks in which they engage and by ongoing reflection on 
their work. The teacher, by observing the students' work and reflecting on 
their learning processes, constructs and understanding, which enables him or 
her to vary the learning environment in ways that are more appropriate for the 
students. Although both the students' learning processes and the interactive 
teaching process are autonomous, the two systems are nevertheless 
interdependent. This interdependence can explain how teachers learn through 
their teaching. 

Zaslavsky et al. (2003) extend this model to teacher educators' learning with 
one more layer. 

Another model of teachers and teacher educators' development, strongly 
emphasising collaboration, is the co-learning partnership described by Jaworski 
(1997) and Bednarz (2000), in which "teachers and educators learn together in a 
reciprocal relationship of a reflexive nature". Margolinas et al. (2005), using a 
model extending Brousseau's work, carry out two case studies of lessons in which 



the teacher's didactical knowledge seemed insufficient for dealing with students' 
solutions and identified two kinds of teacher's learning. 

Let us notice that the three quoted models also implicitly suggest that teachers 
learn more from their practice when they provide their students with a rich 
environment allowing for autonomous reflection. There is in some way a dialectic 
process with actions reciprocal to the usual ones: Students' learning Teachers' 
learning Teacher educators' learning. However, Margolinas et al. (2005) 
suggest that it may not be sufficient for general and stable learning. 


As early as 1993, Ball recognized that the tensions (Cohen, 1990) and the 
dilemmas (Lampert, 1985) with which teachers are confronted, are the 
distinguishing features of teaching and must be treated throughout the teacher's 
career. The importance of the practice context and of the contextual ised 
pedagogical contents has been acknowledged in relation to the restructuring of a 
repertoire of interventions (Brodie, 2000; Bednarz, 2000). Grossman, 
Smagorinsky, and Valencia (1999) use the concept of appropriation to determine 
the process by which a person adopts a pedagogical tool available in a particular 
social environment. The degree of appropriation depended on the congruence 
between the values, the experiences and the goals of the members of this culture. 
The American Institutes for Research recognize the influence of the length of the 
training in the transformation of teaching practices (Garet et al., 2001). This 
component permitted the emphasis on the mathematical and scientific contents and 
also to perceive a greater coherence between the teacher's objectives, the standards 
and the other teachers, and offered the opportunity for active learning. Finally, a 
direct relationship is established between what people know and the way they 
learned the knowledge and the practices. 

To consider these components, a variety of concepts were used: dilemmas and 
tensions (Herbst, 2003; Suurtamm, 2004; Herbst & Chazan, 2003), constraints and 
conditions (Barbe et al., 2005; Robert & Rogalski, 2002; Roditi, 2006); the concept 
of practical rationality (Herbst & Chazan, 2003), and the concept of teachers' 
sensibility to a milieu (DeBlois, 2006). 

Tensions and Dilemmas 

Practising teacher education programs usually require a change of paradigm. 
Consequently, the teachers are often confronted with dilemmas that many studies 
have identified. Brodie (2000), Herbst (2003), and Suurtamm (2004) describe how 
a constructivist approach could create tensions and dilemmas. For example, 
questioning students makes their weaknesses more apparent. Experimenting new 
tasks with students requires paying attention to the explicitly expected product or 
to the new ideas which students may develop while engaged in the activity. The 
research of coherent evaluation methods according to the teaching practices, the 



time to explore, develop and find an appropriate curriculum and evaluation 
resources add to the tensions lived by the teachers. 

These dilemmas concern the coordination to be accomplished between the new 
teaching practices and the contents to cover, the students' rhythm, the type of 
questions formulated by the teacher, the conciliation of norms, and "authentic" 
evaluation. Some conditions make it easier to cope with these dilemmas and 
tensions: collaboration between teachers (collegiality), adequate resources (e.g., 
time), school administration support. 

Practical Rationality 

The concept of practical rationality allows practitioners to use their acquired 
knowledge (knowledge, personal engagement) in a similar or different manner than 
the others and to describe the action in its context (Herbst & Chazan, 2003). This 
way, it becomes possible to predict the dilemmas which the teachers will be 
confronted with when they will engage themselves into actions activating these 
dispositions. Like the disposition network, activated in particular situations, it 
becomes possible to justify the presence of very different teaching styles according 
to the objective characteristics of the position of "mathematics teacher" compared 
with connected practices, but still different (e.g., teaching history, doing 

The concept of didactical and ergonomical twofold approach, developed in 
France by Robert and Rogalski (2002), seems to get closer to the concept of 
practical rationality. However, these authors seem to give more consideration to 
non premeditated actions than to those derived from practical rationality. Different 
components are identified and defined: cognitive, mediative, institutional, social, 
personal (Robert & Rogalski, 2002), and collective (Roditi, 2005, 2006). 

Teachers ' Sensibility 

Rene de Cotret (1999) introduces the following distinction between milieu 
(Brousseau, 1997) and environment (Maturana & Varela, 1994): the environment 
relates to the observer's description of a situation and the milieu relates to the 
student's sensibility to this environment. DeBlois (2006) uses this distinction to 
develop the concept of "sensibility" of teachers to speak of their interpretation of 
this milieu (for the student). She observed teachers that were asked to interpret 
their students' errors in mathematics in order to examine the interpretative process 
and its influence on the choice of teaching strategies in a mathematics class. Four 
types of teachers' sensibilities (teaching, familiarity of the student with the task, 
students' understanding, curriculum and characteristics of the task) were identified. 
At that point, it was possible to recognize a relationship between a variety of 
interpretations of students' productions (attention, extension of students' 
procedures, students' ability, product of an interaction between student and the 
task) and kinds of teaching strategies (method of working, creation of a gap with 
the habits, reconsider exercises or manipulative, play with didactic variables). 



When she analyses her data, Leikin (2006) prefers the notion of awareness to 
explain teachers' choices when they teach. She includes this notion in one of the 
two types of factors: momentary factors (reasoning, noticing and awareness) and 
preliminary factors (knowledge, abilities, and beliefs). These two concepts 
(sensibility and awareness) could help to understand the choice of teachers. 


This review of literature shows how difficult it is to organize the variety of results 
and develop some conceptualisation. Nonetheless, we have identified three main 
issues. First, the change of paradigm proposed in practising teacher education 
poses new teaching problems. Research on mathematics teacher education shows 
the importance of flexibility, depth and connectedness of teachers' mathematical 
knowledge. It also shows how difficult it is for teachers to acquire such flexible 
knowledge and use it to manage students' learning in challenging classroom 
mathematics activities. Thus, it seems important to study teaching in its context 
(e.g., in relation with the students or within the larger social or political context). 

Second, when teachers have to change because of an external constraint, reform, 
or new syllabus, it is often difficult for them, and research shows that whereas 
some changes may occur, they are not always the "wanted" ones. The difficulties 
often stem from the very stable imbrications between teachers' beliefs and 
knowledge, and social and institutional expectations as well as cultural context. 
Previous research had already shown that "isolated" changes are not sufficient to 
guarantee real improvement in practice. Some studies show that there are actually 
teachers who learn alone through their teaching, but these studies and inferences 
from other research lead us to believe that these improvements occur inside the 
teachers' previous "teaching style", for example, their beliefs and choices 
regarding contents and classroom management. Perhaps some progress will come 
from better understanding of the differences between this kind of improvement and 
a deeper transformation of practice: what indicators may help researchers assess 
this difference? A way to enlighten this issue may be to study more deeply how 
novice teachers' practice stabilizes during the first five years of their career. We 
meet here one of perspectives by Leikin (this volume) drawn from her analysis of 
research on education of prospective mathematics teachers. Such studies might 
give us evidence of different ways of improving practice according to the level of 
teaching (primary or secondary) and mixing different kinds of knowledge. 
Moreover, recent studies enable us to think that professional growth takes a long 
time and requires a collective investment and the implementation of specific new 
tasks to be used by students and/or teachers. However, due to the complexity of 
these elements of change and the length of time needed, it becomes even more 
difficult for research to assess progress. 

Third, it seems important to construct concepts or systems capable of taking into 
consideration the variety of teachers' work (planning, analysis, classroom 
interactions, including relations with parents etc.). Perhaps some further 
development of research will occur when researchers are able to elaborate models 



of teachers' professional growth which involve mathematical development blended 
with teaching issues and are adaptable to individual specificity. 


The contribution of Lucie DeBlois was partly supported by Social Sciences and 
Humanities Research Council of Canada 410-2005-0406 and the revision of 
English language by DIDIREM-EA 1547 (Paris). 


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i Marie-Jeanne Perrin-Glorian 
■ Equipe DIDIREM 

Ins ti tut Universitaire de Formation des Maitres, Universite d'Artois 


Lucie DeBlois 
\ Centre de recherche sur V intervention et la reussite scolaire (CRIRES) 
\ Faculte des sciences de {'education, Universite Laval 


\ Aline Robert 

\ Equipe DIDIREM 

\ Institut Universitaire de Formation des Maitres, Universite de Cergy-Pontoise 








Multiple Problems and Multiple Solutions 

In this chapter, I first address four interrelated problems that mathematics teacher 
educators (MTEs) are currently facing in the education of prospective mathematics 
teachers (PMTs): (1) The importance of challenging mathematics and PMTs ' 
limited experience in challenging mathematics; (2) Changing approaches to 
mathematics teaching and the resistance to change in teaching; (3) The need to 
intermingle the different components of teachers' knowledge in teacher education; 
and (4) The difficulty of becoming a member of a community of practice. To 
analyse a variety of solutions that MTEs suggest for these problems when working 
with teams of PMTs, I provide in a second part a comparative analysis of 30 
studies focusing on diverse issues integrated in PMT education programmes. 
Finally, I make connections between the problems outlined in the chapter and the 
solutions discussed in the observed studies. The argument presented in the chapter 
is that even though many solutions are suggested and proven to be effective in 
solving some of the outlined problems, we are still lack of evidence that those 
solutions are ample for preparing PMTs to become members of the communities of 
practice they join. I suggest that combining the various solutions proposed for the 
education of teams of teachers in various studies is necessary in order to prepare 
PMTs to be effective mathematics teachers. 


This chapter is based on two main positions: first, teaching is a complex system that 
includes interrelated and mutually dependent elements such as teachers, students, 
curriculum, textbooks, school management, students' families, local settings, and 
other factors that effect classroom procedures; second, teaching is a cultural 
activity, reflected in knowledge and beliefs that guide behaviour and determine the 
expectations of the participants. The chapter also assumes that prospective teachers 
should be prepared to teach in ways that would allow students to fulfil their 
learning potential, and that the mathematical challenge is an irreducible element of 
mathematics education. Based on these assumptions and on the analysis of the 
research literature, I outlineybwr interrelated problematic issues in the education of 
prospective mathematics teachers. 

K. Kramer and T. Wood(eds.), Participants in Mathematics Teacher Education, 63-88. 
© 2008 Sense Publishers. All rights reserved. 


The Importance of Challenging Mathematics and PMTs ' Limited Experience in 
Challenging Mathematics 

The learners' intellectual potential is a multivariable function of ability, 
motivation, belief, and learning experiences (National Council of Teachers of 
Mathematics, 1995). Principles of "developing education" (Davydov, 1996), which 
integrate Vygotsky's (1978) notion of ZPD (Zone of Proximal Development) and 
Leontiev's (1983) theory of activity, claim that to fulfil the learners' mathematical 
potential the leaning environment must involve challenging mathematics. A 
mathematical challenge is an interesting mathematical difficulty that a person can 
overcome (Leikin, 2007). Mathematical challenge is subjective because it depends 
on the learner's potential. 

The importance of the mathematical challenge and its student-dependence in 
teaching and learning is reflected in Jaworski's Teaching Triad that synthesizes 
three core elements: the management of learning, sensitivity to students, and 
mathematical challenge (Jaworski, 1992, 1994). Brousseau ( 1 997), in his Theory of 
Didactical Situation, claimed that one of the central responsibilities of a teacher is 
devolution of good (challenging) tasks to learners. Both the teaching triad and the 
theory of didactical situation stress that teachers ought to provide each and every 
student with learning opportunities that fit their abilities and motivate their 

Mathematical challenges may appear in different forms in mathematics 
classrooms. These can be proof tasks where solvers must find a proof, definition 
tasks in which learners are required to define concepts, or investigation tasks. One 
way of helping teachers to use challenging mathematics in their classes is to 
provide them with appropriate learning material (e.g., a textbook), making a large 
number of challenging tasks available to them (Barbeau & Talor, 2005); but merely 
providing teachers with ready-to-use challenging mathematics activities is not 
sufficient for their implementation: teachers should be aware of the importance of 
mathematical challenges and convinced about them, and they should feel safe 
(mathematically and pedagogically) when dealing with this type of mathematics 
(Holton et al., in press). Furthermore, teachers should have autonomy in employing 
this type of mathematics in their classes (Krainer, 2001; Jaworski & Gellert, 2003). 
They should be able to choose mathematical tasks themselves create those tasks, 
change them so that they become challenging and stimulating, and naturally they 
must be able to solve these problems. 

But despite the importance of teacher awareness of the role of mathematical 
challenge in teaching and learning, prospective teachers often have limited 
experience in challenging mathematics, and sometimes have strong negative 
feelings about it (Gellert, 1998, 2000). In other cases, PMTs find challenging 
mathematical activities interesting and encouraging, but are not sure whether these 
activities are applicable to students. These views are connected to novice teachers' 
inclinations to rely on their procedural understanding of mathematics when making 
pedagogical decisions about mathematical challenges (Borko et al., 1992) or when 
planning or discussing their ideas for teaching (Berenson et al., 1 997). Many PMTs 



encounter difficulties in coping with challenging mathematics themselves, and 
their beliefs about the nature of mathematical tasks contradict the character of the 
tasks they are asked to solve (Cooney & Wiegel, 2003). 

Changing Approaches to Mathematics Teaching and the Resistance to Change in 

The beginning of the 21st century is full of ever better and more advanced 
technological tools. In a changing world, invention and progress can be anticipated 
and developed only by human beings with rich imagination, deep and broad 
knowledge, and solid proficiency. In a changing world, learning and teaching 
environments, informational resources, interpersonal communications, and the 
roles of teachers and students in the classroom adapt constantly to the latest 
modifications. Mathematics education is a typical example of a subject that 
experiences ongoing, multifaceted change, manifest in the shift toward the 
dynamic and investigative nature of mathematical tasks, toward multiple uses of 
technological tools for teaching and learning, and toward the dialogic learning 
environment (Lagrange, Artigue, Laborde, & Trouche, 2003). 

Inquiry and experimentation are basic characteristics of the development of 
mathematics, science, and technology. Inquiry (experimentation) tasks in 
mathematics classrooms are usually challenging, cognitively demanding, and 
enable highly motivated work by students (e.g., Yerushalmy, Chazan, & Gordon, 
1990). Borba and Villarreal (2005, pp. 75-76) stressed that the "experimentation 
approach gains more power with the use of technological tools" by providing 
learners with the opportunity to propose and test conjectures using multiple 
examples, obtain quick feedback, use multiple representations, and become 
involved in the modelling process. 

Educational technologies such as computers and graphic calculators can be 
viewed as cultural tools that reorganize cognitive processes and transform social 
practices in the classroom (Borba & Villarreal, 2005; Goos, 2005). These tools can 
provide a vehicle for incorporating new teaching roles ranging from the 
authoritative "master" to the collaborative "partner" (Goos, Galbraith, Renshaw, & 
Geiger, 2003) and influence the mathematics curriculum (Wong, 2003). 

As approaches to teaching and learning mathematics change, the nature of 
mathematical discourse and socio-mathematical norms changes as well (e.g., Cobb 
& Bauersfeld, 1995; Wood, 1998; Wells, 1999). The shift toward the dialogic 
nature of learning is grounded mainly in Vygotsky's (1978) theory of meaning 
making as the result of the learners' communicative experiences. This approach to 
learning and teaching is based on theories that go beyond cognitive views of 
learning (e.g., Brousseau, 1997; Cobb & Bauersfeld, 1995; Jaworski, 1994; Lave & 
Wenger, 1991; Steinbring, 1998; Wells, 1999). In this context, teaching can be 
considered as a spiral process that facilitates the students' autonomous learning and 
includes planning of learning opportunities for students, presenting challenging 
tasks, monitoring the students' handling of the tasks, and reflecting on learning and 



As a complex system, teaching is stable and resistant to change (Cogan & 
Schmidt, 1999). As a result, changing approaches to school teaching are beset by 
many pitfalls and difficulties (e.g., Lampert & Ball, 1999; Tirosh & Graeber, 
2003). As is true with any cultural activity, teaching is learned through 
participation in activities involving learning and teaching (Stigler & Hiebert, 
1998). In part, future teachers learned to teach when they were prospective teachers 
with certain perspectives on teaching and learning. When they are challenged by 
new teaching approaches, PMTs are often unenthusiastic and reluctant to adopt 
new practices and express preferences for the teaching methods used by their own 
teachers (Cooney, Shealy, & Arvold, 1998; Hiebert, 1986; Lampert & Ball, 1999). 
Education programmes have a special role in supporting educational reform by 
developing teachers' knowledge and beliefs (Llinares & Kramer, 2006). 

The Need to Intermingle the Different Components of Teachers ' Knowledge in 
Teacher Education 

Teachers' knowledge and beliefs determine their decision making at all stages of 
teaching (Ball & Cohen, 1999; Cooney, et al., 1998; Even & Tirosh, 1995; 
Shulman, 1986; Thompson, 1992). The complexity of teachers' knowledge within 
the context of the complexity of teaching itself is one of the main problems in the 
education of PMTs (Llinares & Krainer, 2006). To demonstrate this complexity, I 
use a 3D model of teacher knowledge (Leikin, 2006) that combines three main 
perspectives adopted by researchers in discussing teachers' knowledge: kinds of 
knowledge (Shulman, 1986), conditions of knowledge (Scheffler, 1965), and 
sources of knowledge (Kennedy, 2002). 

From the perspective of the kinds of teacher knowledge (Shulman, 1986), 
teachers' subject-matter knowledge (SMK) comprises the PMTs' own knowledge 
of mathematics and of the philosophy and history of mathematics. Teachers' 
pedagogical content knowledge (PCK) includes knowledge of how students cope 
with mathematics and knowledge of the appropriate learning setting. Ball and 
Cohen (1999) discussed "mathematics knowledge for teaching" that allows 
teachers to unpack their SMK in order to develop deep and robust mathematical 
knowledge in students. Teachers' curricular content knowledge (CCK) includes 
knowledge of different types of curricula and understanding the different 
approaches to teaching. Under conditions of changing approaches to school 
mathematics, this type of knowledge can endow teachers with flexibility in shifting 
among various curricular approaches. 

Kennedy (2002) classified teachers' knowledge according to the sources of 
knowledge development. Systematic knowledge, she argued, is acquired first 
through personal experiences as school students, then through participation in 
courses for teachers, reading professional literature, and interacting with 
colleagues. Prescriptive knowledge is acquired through institutional policies and is 
manifest in tests, accountability systems, and texts of a diverse nature. In contrast, 
craft knowledge is developed largely through experience. This type of knowledge 
relates to teachers as members in a community of practitioners, and is based mainly 



on teachers' interactions with their students and on teachers' reflections on these 

The distinction between knowledge and beliefs as conditions of teacher knowledge 
started with Scheffler (1965) and has been presented in the works of mathematics 
education researchers (e.g., Thompson, 1992; Cooney et al., 1998). Knowledge has 
operational power whereas beliefs are of propositional nature solely. An individual 
has proofs for the facts that belong to his/her knowledge, whereas beliefs are 
accepted without proof. The additional distinction between formal and intuitive 
knowledge is consistent with the views of Atkinson and Claxton (2000), who 
discussed teachers as intuitive practitioners, and differentiated between teachers' 
intuitive knowledge as determining actions that cannot be premeditated and their 
formal knowledge, which has to do mostly with planned teacher actions. 

Professional development programmes must be consistent with the complex 
structure of teacher knowledge. The complexity of professional development 
programmes lies in searching for a reasonable balance between mathematics and 
pedagogy and in connecting between them (Peressini, Borko, Romagnano, Knuth, 
& Wills, 2004). The balance between systematic and craft modes of development 
is also important because if teachers develop their own mathematical 
understanding in a systematic mode, only practice can persuade them that 
implementation of this type of mathematics in the classroom is valid (Leikin & 
Levav-Waynberg, 2007). Because PMTs lack experience as teachers, pedagogical 
and craft knowledge are among the most challenging issues in the professional 
development of prospective mathematics teachers. 

The Difficulty of Becoming a Member of a Community of Practice 

From the social perspective, teachers are considered to be members of communities 
of practice characterized by common norms, routines, sensibilities, artefacts, and a 
vocabulary that are the result of the situated nature of the teachers' practice (Lave, 
1996; Lave & Wenger, 1991). This practice is embedded in a cultural enterprise 
that is also a complex system of the beliefs about society, educational policies, 
auricular requirements, assessments, and the school environment (e.g., Stigler & 
Hiebert, 1999). Teachers' understanding of mathematics and pedagogy within the 
community of practice is bounded by socially constructed webs of beliefs that 
determine the teachers' perception of what needs to be done (Roth, 1991; Brown, 
Collins, & Duguid, 1989). 

The community of mathematics teachers is usually regarded as one of learners 
who continually reflect on their work and make sense of their history, practice, and 
other experiences (Lave & Wenger, 1991). In other words, teacher knowledge 
develops socially within communities of practice, and in turn determines these 
practices. The situation of PMTs is very different from that of practising teachers. 
When PMTs begin their studies, they are (in the best case) members of a team. 
Krainer (2003) maintains that 



"Teams" (and project groups) are mostly selected by the management, have 
pre-determined goals and therefore rather tight and formal connections within 
the team. [In contrast,] "communities" are regarded as self selecting, their 
members negotiating goals and tasks. People participate because they 
personally identify with the topic, (p. 95) 

PMTs may find that they are simultaneously members of different teams in the 
different courses they attend. One of the purposes of educational programmes is to 
develop the norms, routines, sensibilities, artefacts, and vocabulary that will help 
PMTs join their future professional communities. But PMTs rarely emerge from 
their mathematics teacher preparation programme as members of these 
communities. Moreover, they realize that many teachers work more or less 
individually, some of them collaborate because the department or school 
challenges them to do so; in general, it is rather rare, that teachers really take part 
or form self-selected communities of practice. Furthermore, when starting their 
teaching in a school in which a group of mathematics teachers abide by different 
norms than the ones taught and learned in the programme, they often return to their 
point of departure (e.g., Peressini et al., 2004). PMTs should be prepared for 
integration into the collectives they join - especially into those that adopt ideas 
contrary to those stressed in their teacher education programme. As newcomers 
they must take an active part in advancing those groups towards communities, 
prompting innovations, making communities creative and adapting to changes in 
society and culture. 

In sum, preparatory programmes for PMTs must answer various complex and 
sometimes contradictory questions. They should be aimed at developing new 
generations of teachers ready to teach new generations of students in a changing 
world. They must develop the PMTs' mathematical understanding to enable them 
to approach challenging mathematical tasks successfully, design tasks creatively, 
and be flexible in the implementation of the tasks in their classes. Teacher 
programmes should prepare PMTs to teach in ways that are different from those in 
which they learned as students. They need to prepare PMTs to be effective, 
confident, and creative users of new educational approaches, rich in technological 
tools (Goos, 2005). PMTs should become acquainted with approaches useful for 
studying their own teaching practice as well as that of others, and in analysing the 
effects of teaching on learning. They must aim at generating enthusiasm, intuitions, 
and beliefs about the introduction of challenging mathematics in school in the form 
of reform-oriented pedagogy. 


MTEs offer a wide range of courses and programmes for the professional 
development of PMTs, aimed at solving the problems described above. 
Corresponding research on knowledge, beliefs, and the education of PMTs is 
characterized by the diversity of focal points, types of knowledge involved, level of 



mathematics addressed, and the research tools used in the investigations. In this 
subchapter, I present a review of 30 studies exploring the education of PMTs'. The 
choice of the studies was based on the two main issues: (1) the studies focused on 
the education of prospective teachers as members of teams, (2) they analysed 
development of teachers' knowledge, skills or beliefs rather then examined their 
knowledge as a in its present condition. The selected papers were published 
between 1998 and 2007 in Educational Studies in Mathematics (ESM), the Journal 
of Mathematical Behavior (JMB), and the Journal of Mathematics Teacher 
Education (JMTE). Table 1 summarizes the following characteristics of the 30 

• Type of knowledge (skills, beliefs): The distinction was mainly between 
research focused on SMK and PCK. 

• Level of PMT participants: This column addresses teachers in elementary 
school, secondary school, and other populations involved in the studies. At 
times, the relation between these characteristics is apparent. For example, in 
the first category, studies focusing on SMK were conducted with secondary 
school PMTs, whereas studies focusing on PCK involved primary school 

• Number of participants: most of the studies were performed with one group 
of PMTs who participated in a specific course; several were case studies 
reporting on a small number of participants; and some involved a large 
number of PMTs who completed a research questionnaire. Note that studies 
based on a small number of participants (N=l, 2, 3) were included in the 
review because the selected PMTs were representatives of teams. 

• The setting in which the study was performed: for example, mathematics 
course (MC), didactic course (DC), or teaching practicum (TP). This column 
also includes information about the "mathematics of change" in a computer- 
based environment. 

• Research tools: for example, individual interviews, examples from video- and 
audio-recordings, artefacts of the students' activities, individual journals, and 
written questionnaires. 

• Focal issues of the research: these characteristics are detailed further in this 
subchapter with respect to the findings of the studies under consideration. 

Note, that some of the authors did not report all these characteristics, which 
accounts for empty cells in the table. 

The studies are divided into several categories with respect to the roles PMTs 
play in the research interventions and the balance between systematic and craft 
modes of development implemented in the courses. The categories are not 
mutually exclusive, and some studies can belong to more than one category. 
Moreover, other categorizations are possible (e.g., Cooney & Wiegel, 2003; 
Jaworski & Gellert, 2003), but the focus of the present analysis is on the balance 
between mathematics and pedagogy on one hand, and between systematic and craft 
modes of learning on the other. 



Table 1 



of knowledge 







Research tools 

of the study 

Development through Personal Experiences as&etraera ?- *" v " *•: •* &'£"* , * - 

Applying context-based approaches to course design (CD) 

1 . Cavey & Berenson (2005) 






Case study 

CD, growth in understanding right 
triangle trigonometry 

2. Furinghetti (in press) 








CD, how history affects the construction 
of teaching sequences in algebra 

3. Heaton & Mickelson (2002) 








CD, statistical knowledge, views on 

teaching statistics, project-based 


4. Lavy & Bershadsky (2003) 






(2 lessons) 


Types of participant-generated 
problems, mathematical difficulties 

5. Nicol(2002) 





SMK and 


G-int, I-Int 

PMT journals 

Researcher field 


CD, style of teaching, 

changing PMTs* belief that they must 

reproduce the style of mathematics 

teaching seen in their school days 

6. Philippou & Christou ( 1 998) 








CD, identifying and changing PMT 
attitudes and beliefs about math 

7. Taplin& Chan (2001) 







Group discussion 

CD, student altitudes toward problem- 
based learning and critical pedagogical 

















of the study 

8. Wubbels, Korthagen, & Broekman 








Longitudinal study 

comparing two 


Quest, I-lnt, video 

CD, student and teacher views of 

mathematics and mathematics 

education, more inquiry oriented 


9. Zbiek & Conner (2006) 







1-Int, video, audio, 

CD, how mathematical understandings 

can develop while learners engage in 

modelling tasks 

Socio-mathematical norms and psychological processes 

10. Blanton (2002) 







Observation, video 

CD, notions about mathematical 

11. McNeal & Simon (2000) 







Observation, video 

Mathematical and pedagogical 

development, processes of negotiation 

of norms and practices 

12. Szydlik, Szydlik, & Benson (2003) 







Beliefs about the nature of mathematical 

13. Tsamir (2005) 







Lessons, video, 

CD, awareness of the role of intuitive 
rules, learning math 

14. Tsamir (2007) 






Lessons, video, 

CD, awareness of the role of intuitive 
rules, learning math 

15. Ponte, Oliveira, & Varandas 


and identity 

Attitude test of 

use of 





Intemet and 



CD, awareness of the role of intuitive 
rules, learning math 

















of knowledge 






Research tools 

of the study 

Focusing on the Teaching Process ^ 

Using multimedia cases (MCA) 

16. Doenr & Thomson (2004) 







Questionnaire, I-Int 

observation, field 


CD, use of cases, teacher educators' 

17. Masingila & Doerr (2002) 





Class observation, 
student notes, final 
paper, questionnaire 
instructor journals, 
researcher field 


for using students' thinking in guiding 
classroom experience 

18. McGraw, Lynch, Koc, Budak, & 
Brown (2007) 








Observation, videos 
transcripts of 




Online and face-to-face discussions, 

classroom implementation of tasks, 

task characteristics and appropriateness, 

developing content, and pedagogical 

content knowledge 

19. Morris (2006) 








Written analysis of 

lesson, student 


source of problem 

Ability to collect evidence about 

students* learning, ability to analyse and 

revise instruction 

20. Santagata, Zannoni, & Stigler 









Pre/post-assessmen t 

What do PMTs learn from the analysis 

of videotaped lessons? 

How to measure PMTs' analysis ability 

and its improvement 





of knowledge 






Research tools 

of the study 

Teaching individual students (TIP) 

21. Ambrose (2004) 







I-int: pre/post, 

written work, field 


Changes in beliefs and skills 

22. Bowers &Doerr (2001) 






Math worlds 


Participant works, 
written reflections 
on teaching, daily 
journals (AHA! 

Thinking: as learners (about math with 

computers); as teachers about students' 


23. Crespo (2000) 




DC -Letter 


with 4th 



Journals: reflection 
on activities, 
case reports 

(Changes in) learning about students' 


interpretive practices 

24. Lee (2005) 







math with 


Videos, written 

works (PMTs' and 


Teacher's role in facilitating students' 

math problem solving with 

technological tool 


















of knowledge 







Research tools 

of the study 

Teaching practicum (TP) 

25. Blanton, Berenson, & Norwood 






Case study, 





Supervision in teacher education 

26. Goos(2005) 





4 of 18 





4 case studies 

Working with technology 

27. Nicol(1999) 




14 of 34 


Course video, 

PMT journals, 

TE journals 

Learning to teach 

(what appears to be problematic) 

28. Nicol & Crespo(2006) 



use of 


4 of 33 


Mnt, pre, med, post, 

course work, 

class observation 

Learning to teach 
(use of textbooks and teaching) 

29. Rowland, Huckstep, & Thwaites 






24 videotaped 

Contribution of knowledge to teaching 

30. Walshaw (2004) 






Questionnaire about 

recent leaching 

practice experience, 


Instances or leaching knowledge in 

production, as interpreted by prospective 



Development through Personal Experiences as Learners 

These studies examined the education of PMTs through personal learning 
experiences and reflection on those experiences. These studies assumed that PMTs 
should be involved in authentic mathematical activities in order to develop SMK 
and PCK, to advance the understanding of constructivist socio-mathematical 
norms, of the uses of technology in mathematics education, and of awareness of 
psychological issues in the teaching and learning of mathematics. Studies in this 
category analysed PMT development through participation in mathematical, 
didactic, or psychological courses that integrated different approaches to teaching 
and learning mathematics. Courses in this category can be subdivided into two 
groups: applying context-based approaches to course design and focusing socio- 
mathematical norms and psychological processes. 

Applying context-based approaches to course design. Based on the assumption that 
PMTs need a context that would allow them to look in a different way at the topics 
they will be teaching, the studies in this group suggest a context-based course 
design. Most of the studies were carried out within the framework of didactics of 
mathematics courses. Among the contexts that were proposed for the course 
design, we find the history of mathematics (Furinghetti, in press; Philippou & 
Christou, 1998), project-based learning within a statistical context (Heaton & 
Mickelson, 2002), realistic mathematics (Wubbels, Korthagen, & Broekman, 
1997), mathematical modelling (Zbiek & Conner, 2006), right triangle 
trigonometry (Cavey & Berenson, 2005), mathematics in a workplace (Nicol, 
2002), and problem-based learning (Taplin & Chan, 2001). 

These studies demonstrated that a context-based design of courses for PMTs 
was effective when using content previously unknown to the PMTs. For example, 
in Zbiek and Conner's study the objectives of the course included both learning 
mathematical modelling and learning to develop and implement application 
problems and mathematical modelling tasks in future classrooms. They found that 
the course opened opportunities for PMTs to grow as "knowers and doers" of 
curricular mathematics. Furinghetti analysed how history affected the construction 
of teaching sequences in algebra based on activities carried out at the "Laboratory 
of Mathematics Education". The aim of the course was to equip PMTs with 
understanding of the cognitive roots of the concepts and processes that their future 
students were going to encounter in algebra. The study showed that the integration 
of history in school mathematics inspired variability in strategies of teaching, and 
that the fact that students had not had specific preparation in the history of 
mathematics opened diverse opportunities for the development of prospective 
teachers' SMK and PCK. 

Overall, these studies demonstrated that as a result of systematic implementation 
of the suggested context-based approaches in the courses for teams of prospective 
teachers the changes that took place in the PMTs' knowledge and beliefs occurred 



both in the field of mathematics and pedagogy. At the same time, these studies did 
not ask whether these learning experiences were powerful enough to equip PMTs 
with norms that will help PMTs join their future professional communities. 

Socio-mathemalical norms and psychological processes. The studies in this group 
acknowledged the importance of involving PMTs in authentic activities focusing 
particular socio-mathematical norms. The motivation of these studies was to design 
undergraduate experiences organized around reform-minded ways of teaching in 
order to close the gap between reform-oriented mathematics and the PMTs 
previous mathematical experiences and conceptions about mathematics and the 
teaching of mathematics. Two main modes of course design can be observed here. 
The first one - a mathematical mode - in which PMTs' learning through 
challenging mathematical experiences leads to the development of both SMK and 
PCK (e.g., Blanton, 2002; McNeal & Simon, 2000). The second one - a didactic 
mode - in which MTEs emphasize changes in approaches to teaching mathematics, 
leading to the development of both PCK and SMK through PMTs' experiences 
with innovative pedagogy (Szydlik, Szydlik, & Benson, 2002). 

The results of these studies show that changes in socio-mathematical norms in 
PMTs' courses influence their conceptions of mathematics teaching. Blanton 
(2002) showed that the undergraduate mathematics classroom offers a powerful 
framework for PMTs to practice, articulate, and collectively reflect on reform- 
minded ways of teaching. The study demonstrated that participants construct an 
image of discourse as an active collective process by which students build 
mathematical understanding and develop their ability to participate in such 
discourse. McNeal and Simon (2000) noted that, in the beginning, most 
prospective teachers were uncomfortable with the mathematics of the course both 
as learners and as future teachers. They argued that the constitution of a classroom 
micro-culture supports knowledge development and demonstrated how through 
participation in the course, students develop a new relationship with mathematics. 
Examining the processes by which PMTs negotiated norms and practices, the 
researchers identified and elaborated categories of interaction central to the 
ongoing negotiation of new norms and practises, and illustrated how each of these 
categories of interaction contributed to the process of negotiation. Szydlik, Szydlik, 
and Benson, (2002) found that participants' beliefs became more supportive of 
autonomous student behaviours. Participating PMTs attributed their changes in 
beliefs to classroom norms that included mathematical explorations, expanding 
problem-solving methods, and the requirement for explanation and argumentation. 
All these studies stressed once again that cognitive development requires a social 
context in which mathematical activities support such development (McNeal & 
Simon, 2000). 

In contrast with the above studies, Tsamir's (2005, 2007) studies examined 
courses from a psychological perspective. Tsamir addressed the accumulating 
knowledge of secondary school PMTs in a course from the psychological point of 
view of mathematics education, explicitly including the intuitive rules theory. 



Tsamir demonstrated that emphasis on the psychological processes involved in 
mathematics learning and teaching allowed advancing PMTs' knowledge of both 
SMK and PCK types. 

The mutual relationships between mathematics and pedagogy (including the 
didactic and psychological aspects) appear clearly in the studies of this category. 
Regardless of whether PMTs participate in mathematical, didactic, or psychology 
courses, both SMK and PCK are developed at once. It may suggested that by 
attending to the two types of knowledge explicitly, MTEs can develop both PMTs' 
knowledge and their awareness of the importance of mathematical challenges and 
pedagogical approaches in teaching and learning mathematics. 

As members of teams of PMTs' the participants of the observed studies 
advanced in their views on socio-mathematical norms, changed their attitudes 
towards "different mathematics" or "new pedagogy" and sometimes were shown to 
develop their mathematical expertise. Moreover, the learning teams when involved 
in challenging learning experiences transformed into communities of learners. 
Through the participation in "unusual" mathematical activities prospective teachers 
developed shared values, norms, routines, appreciation of the role of mathematical 
discussion for the development of mathematical understanding (e.g., Blanton, 
2002; McNeal & Simon, 2000). Still, the question of PMTs' readiness to join their 
future communities of practice was not addressed. Additionally, Santagata, 
Zannoni, and Stigler (2007) found PMTs often found innovative teaching 
approaches to be too abstract and unrealistic (e.g., MTEs often hear PMTs saying: 
"fVe learned the other way, and why do we need this?" or "These experiences are 
good for us as fixture teachers but not for our future students "). 

Thus, the content of the courses should be well connected to the classroom 
context in which PMTs are going to apply that knowledge. It is a common belief 
that in order to bridge the gap between the PMTs' previous experience and the 
desired outcome of their education, programmes for them must include field 
experiences where prospective teachers are exposed to the complexity of the 
classroom and to the reality of having to implement alternative approaches. 

Focusing on the Teaching Process 

Prospective mathematics teachers must be offered more authentic teaching-related 
experiences to prepare them for the complexity and challenges of the school 
context (e.g., Darling-Hammond, 1997). The studies in this category explored 
courses in which PMTs were involved in analysing teaching processes. Courses 
for PMTs vary in the way in which they enable PMTs to learn form experience, 
whether they are based on others' teaching experiences (exposure to examples of 
teaching) or on one's own: (a) using multimedia cases, (b) teaching individual 
students, or (c) making teaching practice an integral part of each educational 
programme for PMTs. The use of video cases may be considered a transition from 
pure learning of mathematics and pedagogy to learning from the teaching 
experiences of other teachers. These studies maintain that observation and 
systematic analysis of video cases and videotaped lessons are effective professional 



development tools. Another group of studies deals with the teaching of individual 
students and reflection on those experiences within a team and can be considered 
as an intermediate stage between systematic and craft modes of development. 
Although PMTs are not in a classroom, they interact with students and learn from 
these interactions. The last category is that of teaching practicum. These studies 
examine the PMTs' involvement in school teaching as individuals with sequencing 
discussion of the teaching experiences in a team. The authors analyse the process 
of learning to teach, supervision in the course of practicum, and the contribution of 
knowledge to teaching. 

Using multimedia cases. Research on video cases continues earlier research that 
examined the use of text-based cases in teacher education (e.g., Barnett, 1991; 
Shulman, 1992; Stein, Smith, Henningsen, & Silver, 2000). The rationale for the 
implementation of video cases includes several considerations. First, it is difficult 
to find sufficient high-quality classrooms for placements; a careful choice of video 
cases can expose PMTs to good teaching. Second, video cases or complete 
videotaped lessons serve as a basis for group discussion, the development of shared 
norms, and reflective thinking on the students' mathematical thinking (see also 
Seago, this volume). Third, videotaped classroom episodes or whole-class 
procedures can develop PMTs' critical evaluation of classroom practice. Fourth, 
videos can be played repeatedly to enable a depth of reflection and analysis that are 
often impossible to achieve in live observations. Overall, the various uses of videos 
allow teacher education programmes to face the challenge of developing PMTs' 
conceptual understanding of how students understand subject matter and how it 
should be introduced to them (Hiebert, Gallimore, & Stigler, 2002). 

Studies on the implementation of multimedia cases demonstrated the following 
outcomes: (I) PMTs were able to learn from video cases about students' learning, 
to analyse the effects of instruction, and to revise the initial instruction (McGraw et 
al., 2007); (2) providing ways to measure PMTs' ability to analyse video cases and 
the improvement of this ability along the course (Santagata, Zannoni, & Stigler, 
2007); (3) characterizing ways in which video-cases are used by MTEs and the 
relationship between the PMTs' background and experiences and their uses of 
video cases (Doerr & Thomson, 2004); (4) demonstrating the potential of the case 
studies for developing PMTs' ability to analyse critically classroom episodes 
(Masingila & Doerr, 2002). 

These studies showed that video cases have a positive effect on the development 
of PMTs' knowledge and skills and identified the complexity of diverse processes 
involved in this development. They stressed once again the complexity of noticing 
(in the sense of Mason, 2002) and the diversity of the focuses of attention: the 
mathematics of the tasks, the interaction between teacher and students, and the 
issues related to whole-class discussions (McGraw et al., 2007; Morris, 2006). The 
studies showed that PMTs analyse dilemmas and tensions revealed in teaching 
recorded in video-cases based on their own perspectives on teaching (Masingila & 
Doerr, 2002). 



Overall, video-case studies may be seen as a preparatory stage in PMTs' 
experiences of analysing teaching practice. I suggest considering video-case 
courses as exemplifying an experimental mode of professional development. By 
analysing other teachers' craft experiences in laboratory conditions PMTs develop 
tools and skills for analysing their own teaching practice. The analysis of video- 
cases performed in systematic mode, under the guidance of MTEs, leads future 
teachers to important inferences about the nature of teacher-student interactions 
and their role in the students' knowledge development. These experiences evolved 
PMTs' understanding of the complexity of teaching. As members of teams, 
through analysis of the video-cases, PMTs developed sensibilities and a vocabulary 
that should prepare them to future teaching practice, and help them to enter there 
future communities (e.g., Morris, 2006; Doerr & Thomson, 2004). 

It should be noted that video-case studies analysed primarily the impact of the 
use of video-cases on PMTs' knowledge of pedagogy. In these studies, little 
attention has been paid to PMTs' learning of mathematics. Additionally, the 
question of whether in their personal teaching interactions with students PMTs will 
be able to use the knowledge and skills advanced through the analysis of video- 
cases remains open until PMTs experience the teaching settings in person. 

Teaching individual students. As an intermediate stage between video-cases and 
teaching practicum, mathematics educators incorporate the teaching of individual 
students, complemented by reflective analysis of their experiences within teams. A 
diversity of focal issues appears in this group of studies. For example, Bowers and 
Doerr (2001) and Lee (2005) analysed secondary PMTs' thinking about the 
"mathematics of change" in a computer-based environment, working individually 
with young children. Crespo (2000) explored PMTs' learning about student 
thinking by analysing their interpretations of the students' works by PMTs engaged 
in interactive mathematics letter exchanges with fourth-grade students. 

A variety of findings associated with learning through teaching individual 
students have been reported. PMTs were surprised to find mathematics teaching to 
be more difficult than they had thought, and inferred that providing children with 
time to think when solving mathematical problems was an irreducible component 
of teaching (Ambrose, 2004). PMTs' foci of attention changed as their roles 
changed: as students they were curious about reconceptualising the mathematical 
theorem they learned, whereas as teachers they built on their students' explanations 
and on the role of the technological tools (Bowers & Doerr, 2001). PMTs tended to 
use their problem-solving approaches in their pedagogical decisions: they asked 
questions that would guide students to their own solution strategies; recognized 
their struggle in facilitating students' problem solving, and focused on improving 
their interactions with students. PMTs used technological representations to 
promote students' mathematical thinking, and used technological tools in ways 
consistent with the nature of their interactions with students (Lee, 2005). PMTs' 
interpretations of students' works developed in the course of their experiences: in 
the beginning, PMTs attended to the correctness of students' answers and later they 



focused on meaning; they started from quick and conclusive evaluation and shifted 
to a more complex, thoughtful, and tentative approach. 

Overall, the studies that analysed PMTs' teaching experiences with individual 
students demonstrated advance in PMTs' understanding of the teaching process. 
Instructing individual students helped break the well-known conviction loop: to 
implement new pedagogical approaches, teachers must be convinced of the 
suitability of those approaches in their work with students and, at the same time, to 
be convinced of the suitability of those approaches they have to implement them in 
school. Experiencing teaching with individual students allowed PMTs to feel more 
confident and to gain those convincing experiences. By discussing these 
experiences with their team-mates, PMTs realized they all had common 
difficulties, surprises, unexpected events and satisfaction. Did PMTs, only when 
teaching, begin to understand that teaching was complex and required different 
levels of attention? Only in craft mode PMTs started feeling what it meant to be 
flexible in teaching and sensitive to the students. The role of teams in these courses 
was to enhance PMT's reflective skills, to provide them with mutual support, and 
sharpen their critical reasoning. 

Teaching practicum. Teaching practicum is one of the professional development 
settings that enable PMTs to make the connection between learning and teaching, 
but requires negotiation between the school and the college or university culture. 
Practicum is a course in which the knowledge of content and pedagogy learned in 
systematic mode is implemented, and craft knowledge is developed almost for the 
first time in the PMTs' professional career. The teachers find themselves teaching 
individually school students and then discussing those experiences with supervisors 
or team-mates. I suggest that courses of this type are representative of another 
mode of professional development: implementation mode. 

Several studies performed in the last decade analysed the effects and 
characteristics of diverse components of teaching practicum on the professional 
development of PMTs. These studies explored the role of university supervision 
(Blanton, Berenson, & Norwood, 2001); pedagogical practices and beliefs about 
integrating technology into the teaching of mathematics (Goos, 2005); ways in 
which PMTs employ their knowledge of mathematics and pedagogy in their 
teaching (Rowland et al., 2005); issues that PMTs find problematic in teaching 
mathematics and changes in the types of questions PMTs ask students (Nicol, 
1999); and the use of textbooks in learning to teach mathematics (Nicol & Crespo, 
2006). Nicol (1999) showed that PMTs - as a result of experiences gained in 
practicum - began to consider students' thinking and to create spaces for inquiry 
through the types of questions they posed. PMTs also began to see and hear 
possibilities for mathematical exploration that evolved as their relationship with 
mathematics and students changed. Nicol and Crespo (2006) showed that PMTs' 
attempts to modify textbook lessons posed pedagogical, curricular, and 
mathematical questions that were not easily answered by reference to textbooks or 



teacher's guides. Findings indicated that practicum could challenge prospective 
teachers to be creative and flexible users of curriculum materials. 

In this group, we also find studies that constructed theoretical models and tools 
(e.g., Goos, 2005; Rowland et al., 2005). Goos (2005) theorized PMTs' learning 
using the notions of Zone of Proximal Development (ZPD), Zone of Free 
Movement (ZFM, possible teaching actions), and Zone of Promoted Action (ZPA, 
the efforts of a teacher educator that are needed to promote particular teaching 
skills or approaches). Using these concepts and the mutual relationships between 
them, Goos demonstrated a variety of relationships between a range of personal 
and contextual factors that influence the formation of the PMTs' identity as 

The study by Rowland et al. (2005) proposed a set of four units as a framework 
for lesson observation and mathematics teaching development. The four units 
were: foundation, transformation, connection, and contingency. Foundation refers 
to teachers' awareness of purpose, the theoretical underpinning of learning and 
pedagogy, and the use of various tools. Transformation refers to knowledge-in- 
action as revealed in manner in which the teacher's own meanings are transformed 
to enable students to learn, including the use of analogies and examples. The 
teachers perform connections between different meanings and descriptions of 
particular concepts or between alternative ways of representing concepts and 
carrying out procedures. Contingency is revealed by the ability of the teacher to 
respond appropriately to contributions by students during a teaching episode. 

Overall, the studies on teaching practicum involved various issues and 
participants in the process of learning-to-teach. The studies analysed primarily 
PMT's pedagogical skills and beliefs, including their understanding of students 
(and their errors), teaching with technological tools, and the use of textbooks. Note 
that these studies paid little attention to the development of SMK in the process of 
teaching or to the extent to which PMTs implemented in practice the material that 
had been studied in systematic mode. The role of the teams in the teaching 
practicum courses was especially important for the development of PMTs' 
reflective and analytical skills (e.g., Nicol, 1999). The prospective teachers when 
discussing their teaching experiences were exposed to the variety of views on 
teaching profession, could reify their own position, learn from own and others' 
"mistakes". Teaching practicum got PMTs closer to the communities of practice 
that they would join in near future. 


MTEs face inherent dilemmas and challenges when preparing teachers for work in 
classrooms. The studies described above demonstrated the diversity of solutions 
that MTEs use in order to solve various issues in teacher education and to support 
prospective mathematics teachers' conceptual changes. In a majority of the studies 
under consideration, these solutions took the form of various professional 
development tools that MTEs integrated in the courses designed for PMTs. Their 
effectiveness was shown by analysing changes in PMTs' knowledge, beliefs, and 



attitudes. Other studies designed theoretical models and tools that may be effective 
in analysing and describing changes in teachers' knowledge and beliefs. In this 
subchapter, I return to the problems highlighted in the first subchapter and outline 
further research questions associated with the education of PMTs. 

Attending to the Centrality of the Mathematical Challenge 

Most of the studies that examined PMTs' development through their personal 
learning experiences included mathematical challenge among those experiences 
(e.g., McNeal & Simon, 2000; Taplin & Chan, 2001; Zbiek & Conner, 2006). 
Little attention has been paid, however, to the concept of mathematical challenge 
itself, although this meta-mathematical awareness is complex and crucial for the 
ability to design and analyse a lesson effectively (e.g., Holton et al., in press). I 
suggest that the notion of "mathematical challenge" as a meta-mathematical and 
psychological concept may serve as a springboard for the development of PMTs' 
knowledge and beliefs. The following questions are important for advancing their 
beliefs in the importance of the mathematical challenge: 

What are the PMTs' conceptions of the mathematical challenge and its role in 
teaching and learning mathematics? How can programmes for teams of PMTs 
foster these conceptions so that PMTs would be eager to implement them in their 
future practice? How can these programmes promote PMT's expertise in solving 
challenging mathematical tasks and their capability of choosing and designing 
those tasks for teaching? 

Attending to Changing Approaches in Mathematics Teaching and Learning 

PMTs' mathematical knowledge and beliefs about mathematics and about teaching 
mathematics are influenced significantly by their experiences in learning 
mathematics long before they decided to become teachers (Cooney et al., 1998). 
PMTs bring these experiences to their teacher education programmes in the form 
of conceptions, and are expected to make changes in their views of mathematics 
and pedagogy. I suggest that the ideas of conceptual change theory found in 
science education (Posner, Strike, Hewson, & Gertzog, 1982) may be useful in 
addressing this issue. Conceptual change acknowledges the importance of prior 
knowledge to learning and considers both the enrichment of existing cognitive 
structures and their substantial reorganization (Schnotz, Vosniadou, & Carretero, 
1999). Such reorganization is conceptualised as being motivated by dissatisfaction 
with the initial conception and by the intelligibility, plausibility, and fruitfulness of 
the new conception (Posner et al., 1982). When teaching PMTs about learning 
alternative approaches to teaching mathematics, MTEs must reconceptualise their 
initial views on teaching and learning. 

Most of the studies reviewed in this chapter showed the effectiveness of 
integrating alternative approaches to mathematics teaching and learning within 
courses or programmes for PMTs. They argued that that programmes focused on 
alternative approaches to teaching and learning mathematics must require from 



PMTs to experiment with these approaches with their students (Hiebert, Morris, & 
Glass, 2003), and to analyse and discuss these experiments (e.g., Bowers & Doerr, 
2001; Crespo, 2000; Lee, 2005; Goos, 2005; Nicol, 1999). But in order to be able 
to experiment the alternative approaches, PMTs must encounter conceptual change 
in the field of pedagogy or of mathematics. Thus, beyond what has been achieved 
already in the reviewed studies, the following questions can be raised to further 
analyse and strengthen PMTs' learning in the various university and college 

What are the ways in which PMTs can achieve dissatisfaction with their initial 
conceptions of school mathematics and mathematics teaching? How can courses 
for teams of PMTs advance their perception of the new approaches as intelligible, 
plausible, and fruitful for teaching? 

Attending to the Complexity of PMTs ' Knowledge 

Most of the studies described course design and examined its effectiveness. The 
courses differed in the balance between systematic modes of development (through 
learning) and craft modes (through teaching), between knowledge and beliefs, 
between mathematics and pedagogy. From this perspective, and together with the 
analysis of the research provided in the second subchapter, I suggest that there are 
four main modes of professional development that vary with respect to explicit 
versus implicit goals, the balance between mathematics and pedagogy, the role of 
challenging content, and the mechanisms of teachers' knowledge development. 
These are the mathematical, pedagogical (didactic or psychological), experimental, 
and implementation modes. 

Combinations of different modes can be achieved by PMTs' participation in the 
courses belonging to different modes within a programme, or through the 
integration of different modes in one particular course for teachers. For example, 
such integration is present in studies that combine experimental and mathematical 
modes (Ambrose, 2004; Bowers & Doerr, 2001), mathematical and 
implementation modes (Nicol, 1999; Nicol & Crespo, 2006), or pedagogical and 
implementation modes (Masingila & Doerr, 2002). Balancing between craft and 
systematic modes is especially important: whereas systematic knowledge provides 
a stable base for the development of craft knowledge, only craft knowledge 
contains the necessary convictions and beliefs about the applicability of what has 
been learned systematically. The balance between systematic and craft modes can 
help solve the "conviction loop" (see above). 

Answering the following questions would further advance the design of the 
programmes for the professional development of PMTs: 

What combinations of systematic and craft modes of development are most 
effective in the courses for PMTs? What combinations of mathematics and 
pedagogy are the most effective in the preparation of PMTs? How can the different 
modes be integrated so that they support each other? 



Becoming a Member of a Community of Practice 

All the studies reviewed in this chapter explicitly or implicitly acknowledged the 
importance of integrating newcomers in the school system. Llinares and Kramer 
(2006) and Peressini et al. (2004) stressed that the process of recontextualization of 
what has been learned in teacher education programmes for prospective teachers 
into what will be taught in the classroom is extremely important. There is not 
enough evidence that the professional development of PMTs prepares them for 
such integration and recontextualization. And there is not enough evidence that 
PMTs who changed their views about approaches to mathematics teaching and 
learning will implement these approaches in their future classes. 

When they first begin to teach school, teachers usually learn from more 
experienced teachers. At the same time, experienced teachers can learn from the 
newcomers how to change mathematics teaching and learning so that it fits better 
the new reality, new technologies, and the latest cultural artefacts and advances. It 
is unclear, however, to what extent future teachers are ready to be adaptive agents 
of these approaches. When beginning their teaching careers, new teachers grasp the 
conflict between systematic knowledge (constructed in the teacher education 
programme) and the prescriptive knowledge they develop within the school 
system. They usually find it hard to cope with the complexity of the system, and 
with the gap between what has been learned in teacher education and the reality of 
the school. Longitudinal studies that could answer the following question may 
further advance teacher education programmes: 

What are the springboards and pitfalls in the transitions from systematic to 
experimental learning, from experimental learning to teaching practicum, and from 
teaching practicum to the real classroom? How, when educated in teams, PMTs 
may be prepared to become members of communities of practices? 

Gaining a better understanding of the factors that promote and hamper the 
professional development of PMTs will support MTEs in their complex task of 
preparing their students for teaching careers at all grade levels. 


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Roza Leikin 
Faculty of Education 
University of Haifa 




Developing Teacher Professionals 

The focus of this chapter is on teams of practising teachers brought together in 
formally arranged situations organized by management, such as, subject 
coordinators, school-based or district-based administrators. The chapter begins 
with a description of the goals for developing teacher professionals. I then 
illustrate the current practice of professional development of the past ten years 
with examples from several countries. Common aspects are evident in the structure 
of the professional development programmes. A case study brings the issues of the 
complexity of studying practising teachers into focus. Finally, I use these examples 
to explicate the message about teams and relevant environment, framed in terms of 
inter-dependence and the co-constructed context. 


Current reform initiatives mainly initiated by the National Council of Teachers of 
Mathematics (1989) call for changes in the "core dimensions of instruction" 
(Spillane, 1999; S pi I lane & Zeuli, 1999). Mathematics teachers are expected to 
establish in their classrooms "communities of learners" where students explore 
mathematics in depth and teachers facilitate students' mathematical learning. 
Students are expected to construct mathematics for themselves and develop a 
means of determining the appropriateness of solutions and procedures based on 
arguments used to justify the solutions and procedures. This vision of mathematics 
learners suggests students make meaning of mathematics and comprises a 
corresponding emphasis on achievement for all students (Stein, Silver, & Smith, 
1998; Tirosh & Graeber, 2003). This theme of more learner-centred and 
conceptually-focused instructional practice is international in scope with 
mathematics progressively seen as a critical competency for greater numbers of 
more diverse students (Adler, 2000; Adler, Ball, Krainer, Lin, & Novotna, 2005). 

A common goal of teacher development programmes, therefore, is to develop 
support for student thinking by helping teachers develop interactive and dialogic 
contexts for learning and promoting students' thinking with appropriate questions 
and statements (Sowder, 2007). Teachers' learning to teach for student 
understanding must integrate knowledge of: mathematics content (including 
concepts, processes, and methods of inquiry); student thinking (understanding the 
ways in which students thinking could develop); and instructional practice (nature 
and effects of their teaching) (Carpenter, Blanton, Cobb, Franke, Kaput, & 

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 89-109. 
© 2008 Sense Publishers. All rights reserved. 


McClain, 2004; Jaworski & Wood, 1999). Teachers, often having only been 
participants in a classroom with a traditional or an instrumental approach to 
mathematics teaching, are expected to teach in ways they themselves have not 
experienced. Therefore, many professional development programmes have as a 
goal assisting teachers in paradigmatic shifts in epistemology and instructional 
practice. The goals of supporting changes in teachers' practice, philosophy, and 
beliefs is fundamentally based in the notion that changed practice, raised 
awareness, and changes in beliefs about mathematics, mathematics teaching, and 
learning can result in a more effective environment for student learning (Jaworski 
& Wood, 1999). 

This chapter focuses on professional development experiences for teams of 
practising mathematics teachers; in particular, teams are mostly selected by 
management, with pre-determined goals, which therefore create rather tight and 
formal connections within the team (Krainer, 2003). Although teams may develop 
into collaborative communities, they are not inherently so. Teams of practising 
mathematics teachers may consist of school-based groups or teachers drawn from 
across a school district or region. Subject-coordinators, school-based or district- 
based administrators are examples of management that bring a team together 
within a formal structure. Teacher educators' and management's choice of what to 
offer or require is shaped by a perception of teachers' needs, by the larger context 
in which schooling occurs, and resources (human, social, and capital) committed to 
such endeavours (Borasi & Fonzi, 2003; Nickerson & Brown, 2008). Individual 
teacher participant's decision to participate is affected by factors such as his or her 
perception of meaningfulness, feasibility, work demands, management and 
collegial support (Kwakman, 2003). Although teams of mathematics teachers are 
mostly selected by management for work structured towards predetermined goals, 
in practice, the goals and structure of the work are equally affected by teachers' 
and managements' participation and the design of successful professional 
development can and should evolve and change (Loucks-Horsley, Hewson, Love, 
& Stiles, 1998). As such, the professional development programmes in which 
teams of teachers are engaged is the joint construction of the teacher educators who 
design it, administrators and management who decide what to offer or require, and 
the teachers who choose the programmes and the manner in which they agree to 
participate (Borasi & Fonzi, 2003). In this same sense, context is not deterministic 
but interactively constructed among the participants (Jones, 1997). 

Therefore, the mathematics teacher professional development programmes as 
constituted are reflective of the specific political and cultural contexts in which 
they are embedded. Cooney and Krainer (1996) describe how the nature of the 
programmes for practising mathematics teachers is constructed from macro 
problems and micro problems. Macro problems emanate from society in general 
and are related to economics, politics, culture, and language. Micro problems are 
directly related to problems within mathematics teacher education, such as 
curricula or teacher training. The design of professional development programmes 
is reflective of an attempt to address both macro and micro problems. 



For example, national initiatives affect policy and privilege some foci in the 
distribution of resources. Mathematics teachers' professional development in many 
Western countries is motivated by having not compared well to other countries in 
international comparisons of students' mathematics achievement in the 1990s. 
Consequently, the governments, for example, New Zealand, Australia, some 
European countries and the United States responded to such rankings with an 
impetus to improve student achievement (see e.g., Bobis et al., 2005; Borasi, Fonzi, 
Smith, & Rose, 1999; Keitel & Kilpatrick, 1999). Some countries have a focus on 
repairing inadequate prospective teachers' preparation, which includes the 
mathematical knowledge of teachers and, in the case of South Africa, extends to 
reconstruction of identity (Adler & Davis, 2006). In other countries, mathematics 
education and consequently professional development programmes for teachers are 
shaped by the goal of eradicating economic and technological disparity (Atweh & 
Clarkson, 2001; Tirosh & Graeber, 2003). Thus, teams of teachers are, by 
definition, engaged in professional development structured as top-down 
implementations. As such, political, social, and economic concerns contribute to 
the structure of mathematics professional development for teams of teachers. 

The nature of mathematics teacher professional development is also shaped by 
the mathematics education community's beliefs about mathematics, mathematics 
teaching, and learning. For example, Ball (1997) discusses mathematics education 
reform in the United States as based on concern about students' achievement in 
mathematics and current economic, political, and social pressures for greater 
numbers of students who can use mathematics competently. Yet, the reform is also 
shaped by our ideas of what constitutes mathematics learning and knowledge. 
Constructivist learning theories permeate new directions for teacher professional 
development experiences (Ball, 1997). In many teacher education programmes 
around the world, teachers are seen as constructors of knowledge with a need for 
opportunities for reflection in order to learn (Cooney & Krainer, 1996). Teacher 
educators of practising teachers acknowledge the challenge of paradigmatic 
changes in instructional practice, the need to make connections between 
professional development and work on site, and the benefits to teachers of 
substantial support from colleagues (see e.g., Knight, 2002; Putnam & Borko, 
2000; Tirosh & Graeber, 2003). 

What follows is a survey of studies of professional development programmes 
and approaches that illustrate current themes of mathematics teacher professional 
development for teams of teachers in many countries. Approaches to professional 
development are frequently characterized according to the content of the 
workshops or focus of the work with teachers (see e.g., Borasi & Fonzi, 2003; 
Kilpatrick, Swafford, & Findell, 2001; Sowder, 2007). Most studies of teacher 
development adopt an individual teacher as the unit of analysis, describing changes 
in an individual teacher's knowledge of mathematics, beliefs, or instructional 
practice (e.g., Cohen, 1990; Sowder & Schappelle, 1995). From another 
perspective, professional development is a social matter, enhancing collective 
capability (Knight, 2002). This perspective on teacher development situates 
teachers' work in context and views professional development as the building of 



communities of collaborative, reflective practice, wherein teachers are joined with 
colleagues to create effective mathematics environment for their students' learning 
(McClain & Cobb, 2004; Stein, Silver, & Smith, 1998). 

Researchers have identified some of the critical aspects of effective professional 
development. Notable among these is the need to foster collaboration to encourage 
the formation of communities of collegial learners (e.g., Borasi & Fonzi, 2003; 
Wilson & Berne, 1999), to involve administrators and other stakeholders as they 
are critical to arranging resources (e.g., Gamoran, Anderson, & Ashmann, 2003; 
Krainer, 2001), and the need to carefully consider alignment with the 
organizational context in which these teachers work (e.g., McClain & Cobb, 2004; 
Sowder, 2007; Stein & Brown, 1997; Walshaw & Anthony, 2006). In what 
follows, I describe several professional development programmes that illustrate the 
importance of organizational factors and joint activities when teams of practising 
teachers are engaged in professional development (Krainer, 2001). This survey is 
intended to highlight these themes while simultaneously revealing the breadth of 
possible paths. 


Although the structure of the mathematics professional development described here 
varies with regard to length and focus and in the manner in which the goals are 
addressed, the central goal of these programmes is to support practising teachers in 
reorganizing their instructional practice to become more learner-centred and 
conceptually focused. Teachers learning to teach in a manner that supports student 
understanding means that professional development often focuses on engaging 
teachers in doing mathematics, understanding student thinking and the ways in 
which it develops, and scaffolding attempts at changed instructional practice. 

Teacher education programmes have focused on and continue to focus on the 
central aspect of teachers' knowledge of mathematics (Jaworski & Wood, 1999; 
Sowder, 2007). This knowledge of mathematics is framed more broadly than 
understanding concepts. The mathematical knowledge needed by teachers 
encompasses the ability to use knowledge of mathematics to foster effective 
learning for students (Adler & Davis, 2006; Ball & Bass, 2000). One important line 
of research of the last decade concerns the nature of the mathematical knowledge 
needed for teaching, which in turn influences the design of teacher education as 
mathematics teacher educators try to provide opportunities to learn these 
specialized ways of learning and knowing mathematics (Adler & Davis, 2006). 
Thus, the programmes with a mathematical focus encompass concepts, process, 
methods of inquiry, beliefs about mathematics and mathematics learning. In some 
countries, professional development has a focus on initiating an alternative 
conception with regards to mathematics teaching and learning (see e.g., Farah- 
Sarkis, 1999; Mohammad, 2004; Murray, Olivier, & Human, 1999). 

As an example, in the context of South Africa, the practising teachers and the 
administrator placed limits on the time available for professional development. 



Murray et al. (1999) describe a two-day workshop attended by primary (K-3, 5-8 
year olds) mathematics teachers and head of subject upper elementary teachers in 
South Africa. Within this narrow window of time, teacher educators had a goal of 
changing teachers' perceptions of mathematics "[...] and equipping teachers for 
radically different classroom practice" (Murray et al., 1999, p. 33). Given the 
teachers' impoverished preparation consisting of traditional mathematics 
experiences and low self-perception of mathematical ability, the teacher educators 
tried to address, in their workshop, perceptions of how mathematics is learned and 
used, as well as the teachers' perceptions of their own mathematical ability. By 
posing problems that were challenging to teachers and then supporting reflection 
on their experiences as learners, the designers hoped to justify a problem-centred 
approach to teaching and to share information to support teachers in establishing a 
problem-centred classroom. Murray et al. (1999) noted the importance of engaging 
teachers in mathematical activities that suggested different solution strategies and 
were aligned to the syllabus. The format encouraged reflection and connections to 
students' views in similar activities, and the development of a vision of the 
teacher's role in fostering a problem-centred approach to learning mathematics. 

The teachers' evaluations of the workshop suggested that the activities were 
successful in providing a vision of a "starting place" in their own classrooms. 
Teachers who perceived themselves as mathematically weak reported being deeply 
moved by their experience with mathematical sense-making. While the real test of 
the workshop's success was in changed instructional practice, the authors believed 
that the fundamental groundwork was laid. They acknowledged that the challenge 
of changed practice is ultimately highly dependent on factors such as supervisor 
and peer support. The teams of practising teachers were brought together with 
persons who were heads of mathematics in their schools to develop a vision of how 
mathematics is learned with understanding and the role of the teacher in supporting 
this understanding. After participation, the teachers requested more opportunities 
for professional development of longer duration. 

In stark contrast to the time available with practising South African teachers, 
teacher educators in Israel describe an extensive programme for practising middle 
and high school mathematics teachers and mathematics department chairpersons. 
Each round of the Kidumatica programme consisted of full-day weekly meetings 
for three years, and had a goal of raising the level of teachers' content knowledge 
and of promoting collaboration among teachers teaching at different grade levels. 
Fried and Amit (2005) describe a "spiral" activity employed to provide 
opportunities for teachers to see a problem situation developed for different grade 
levels. A single problem situation was illustrated and modified for each of seventh 
through eleventh grades. The teacher educators addressed a perceived problem in 
instructional practice wherein teachers tended to present problems as one- 
dimensional entities that embody a single technique or concept. The team met for a 
portion of the time into large across-grade groups to encourage reflective 
professional conversations. These conversations were not just about a particular 
task, but rather concerned the links between the mathematics for students of 
different grades or achievement levels. 



One of the goals of arranging across-grade groups was to knit together a broad 
mathematics teaching community within an individual school and a region. Fried 
and Amit (2005) stressed the importance of the multiple perspectives brought by a 
broad grade-level range of teachers both to develop a deeper, connected 
understanding of the mathematics and also to develop respectful relationships with 
colleagues across the grade spectrum. The professional development enhanced 
collective capability of the team in understanding the range of mathematics 
connections and challenged the notion that if one understands the concepts at a 
higher level, then one has the mathematical knowledge needed for effectively 
teaching it to lower middle-grade students. 

A number of professional development initiatives are characterized by the 
central goal of making students' thinking the focus of and the impetus for teachers' 
reflection on their own instructional practice (e.g., Fennema et al., 1996; Loucks- 
Horsley et al., 1998; Sowder, 2007; Zaslavsky, Chapman, & Leikin, 2003). 
Teaching mathematics for understanding requires knowledge of students and how 
their mathematical thinking develops. Examining student thinking focuses 
teachers' attention on the consequences of instructional practices and provides 
opportunities to understand the discrepancies between what students understand 
and what teachers wish them to learn (Loucks-Horsley et al., 1998). Teachers can 
then build upon the concepts and skills expressed in student thinking, using it to 
guide instructional decisions. 

Whitenack, Knipping, Novinger, Coutts, and Standifer (2000) report on a team 
of practising primary teachers brought together to learn about students' thinking. 
There were three phases to the programme; in the first phase, the practising 
primary teachers were part of a larger district-wide group invited to hear two 
presentations by pre-eminent mathematics educators who addressed issues of 
teaching and learning mathematics with understanding and creating a learning 
environment for all students to engage in meaningful mathematics. The teacher 
educators sent out a district-wide invitation to all K-2 teachers requesting they 
submit an application to participate in the programme. The professional developers 
worked with a K-7 district mathematics coordinator, who had a rich history with 
some of the teacher participants, to plan sessions (Whitenack, personal 

During the second phase, 27 practising teachers participated in a one-week 
summer institute. Participants viewed videotapes of students being interviewed 
while engaged in mathematical problem solving then discussed the videos in terms 
of students' number development. A mini-case study assignment was designed as a 
culminating activity for the summer institute. In contrast to case-based professional 
development formats wherein teachers analyse and discuss a presented case, 
teachers in the summer institute were asked to work with a partner to develop mini- 
case studies of students' thinking. The videos included examples of six students (5- 
9 years old) solving various problems. For example, teachers could select to 
investigate the strategies used by one child or identify instances in which several 
students using the same strategy. This provided an opportunity for teachers to 
develop hypotheses about the strategies students used and the kind(s) of 



mathematical reasoning such solutions required and conjectures about what it 
means for students to know and do mathematics. 

During the third phase, the project team met with teachers for eight 3-hour 
sessions during the first four months of the school year. These meetings afforded 
opportunities for teachers to share new ideas they were exploring, and to reflect on 
instructional materials that might support students' understanding of place value, 
multiplication, division, and geometrical concepts. The teachers designed and 
conducted interviews with students from their own classes and implemented 
lessons that addressed concepts that had surfaced during interview sessions. The 
mathematics teacher educators worked with the district mathematics coordinator as 
a means to support teachers' learning about students' thinking. Participation in the 
project facilitated peer collaboration focused around an analysis of students' 
thinking, and scaffolded teachers' trying on new lenses for looking at the student 
work by assisting teachers embedded in the work of teaching. 

Another professional development approach involving teacher-teams was also 
based on the belief that increasing teacher's awareness of students' thinking 
contributes to improvement in teaching, however in this case with middle and 
secondary level students. Tirosh, Stavy, and Tsamir (2001) developed a research- 
based seminar to introduce a theory that would assist practising middle school and 
high school teachers in understanding and predicting students' responses to 
mathematical and scientific tasks. During the course of the seminar, teachers were 
first introduced to the Intuitive Rules Theory, used to explain how students react in 
similar ways to mathematical and scientific tasks. Teachers learned about how 
incorrect answers involving comparison and subdivision tasks can be explained by 
this theory, the educational implications of this theory, and then they engaged in 
discussions of teaching by analogy and conflict. Specific examples of research in 
the context of the intuitive rules are discussed. Finally, each teacher was asked to 
select a topic and define research questions either to conduct a micro-study related 
to validating a known or identifying a new intuitive rule, or the development and 
assessment of teaching interventions to counter unproductive intuitive rules. 
Members of the group met weekly to collectively discuss each other's proposals 
and research questions. Several teachers collectively analysed data and presented 
the results to the others. Although the team of practising teachers met with experts 
outside of the teaching community with an initial structured agenda, in the final 
phase they selected an investigation related to the work of their teaching. These 
practising teachers collectively developed a new lens for collaborative inquiry into 
student reasoning. 

Thus far, the programmes discussed illustrate professional development that is 
grounded in and related to the work of teachers' own teaching and involving 
collaboration with outside experts. The programmes are reflective of the growing 
realization of the importance of involvement of heads of department or other 
stakeholders. They illustrate different means of support toward reorganization of 
instructional practice, often by asking teachers to "try out" aspects of practice that 
they are encouraged to "take back" to the classroom. The following programmes I 
describe are more integral to the institutions in which the teachers work. In each of 



these cases, a level of professional development and scaffolded attempts at 
changed instructional practice occur at the level of the school and within the site. 
Two programmes for Australian teachers focused on the use of research-based 
frameworks for young students' number learning in the early years of schooling, an 
assessment interview to profile a child's knowledge, and whole-school approaches 
to professional development. 

The two initiatives, Count Me In Too (CMIT) in New South Wales and the 
Victorian Early Numeracy Research Project (ENRP), had two goals: to help 
teachers understand students' mathematical development and to improve students' 
achievement in mathematics. A key aspect of CMIT was the Learning Framework 
in Number (LFIN) developed by Wright (1994) and based on Les Steffe's 
psychological model of the development of students' counting-based strategies 
(Bobis et al., 2005). Teachers were assisted in using an assessment to profile a 
student's knowledge across the spectrum of key components of LFIN; this profile 
was then be used to guide instruction. Typically, the programme involved a district 
mathematics consultant and a team of three to five teachers from each school with 
the district mathematics consultant assisting in planning. The project expanded 
from a pilot in 1996 in 13 schools to almost 1700 schools by 2003. One initial 
obstacle to implementing CMIT was the misalignment between the programme's 
content and the national syllabus. In 2002, a new syllabus was released which was 
closely aligned with the CMIT project. During this time, the focus in the 
professional development on number extended to include a measurement strand 
and a space (geometry) strand (Bobis et al., 2005). 

With a similar focus on the use of research-based frameworks, the Victorian 
Early Numeracy Research Project (ENRP), which ran for three years from 1 999- 
2002, introduced teachers to a framework of growth points in young students' 
mathematical learning developed by the project leaders (Bobis et al., 2005). This 
included five or six growth points in each strand of Number, Measurement, and 
Space. Similarly, it involved a task-based assessment interview, and a multi-level 
professional development programme aimed at developing a common "lens" 
through which teachers could view students' reasoning in multiple settings. The 
professional development programme involved engagement at national, regional, 
and school levels. 

The ENRP involved approximately 250 teachers from 35 project schools. The 
support for teachers formally occurred on three levels, state, regional and school, 
(Bobis et al., 2005). All 250 teachers from the state of Victoria met with the 
research team each year for five full days spread across the year. The focus of these 
meetings was on understanding the research framework, the interview, as well as 
appropriate classroom strategies, content, and activities to meet the needs of their 
students. At the regional level, teachers gathered on four or five occasions each 
year usually for two hours after school. These meetings brought together three to 
five school "professional learning" teams and were facilitated by a member of the 
university research team. Teams were made up of all Prep-2 teachers (teachers of 
students 5-8 years old) in each school, an early numeracy coordinator, the principal 
and the early year's literacy coordinator in some schools. The meetings focused on 



sharing particular activities or approaches, mathematical content, and an 
articulation of the tasks to be completed before the group met again. Finally, at the 
school or classroom level, the cluster coordinator spent time on teaching, 
observing, and planning. The coordinator at each school conducted regular 
meetings of the "professional learning team" to facilitate communication, maintain 
continuity and focus. 

Researchers examined the nature of the work of the professional learning teams 
by analysing coordinators' "significant event" folio entries in which they reflected 
on current mathematics education issues at their school. Other data sources were 
interviews of the coordinators and surveys administered to principals of the 
participating schools. The professional learning teams varied in size, operation, and 
meeting frequency. Some teams had the same team members and coordinator 
across the three years of the project, others had a transient team and some had a 
different coordinator each year (State of Victoria, 2003). Early in the project, 
professional learning teams were enthusiastic, yet somewhat overwhelmed by the 
requirements of project participation, while the student interviews provided insight 
into students' thinking, the information also proved daunting for teachers. The 
teams became aware of and were initially uncomfortable with the notion that there 
were many different approaches to teaching. Eventually, teams acknowledged 
common goals for students while accepting professional differences in teaching. In 
the beginning of each subsequent year, the enculturation of new team members was 
seen as a major challenge. The further the project progressed, the greater became 
the discrepancy between the "oldtimers" and the "newcomers". The coordinators 
worked within limited release time to address this discrepancy but the strategies 
and success in mediating this varied across sites. However, both principals and 
coordinators noted increased dialogue around mathematics with more willingness 
to share ideas, opinions, and resources. The researchers concluded that the teams 
served an important role in supporting teachers in improving the teaching of 
mathematics (State of Victoria, 2003). 

These two programmes were motivated by national government interest in a 
remedy for poor achievement in mathematics in international comparisons. The 
initiatives drew on research-based learning frameworks that describe a trajectory or 
pathway of students' early mathematical learning. The frameworks and assessment 
tools provided teachers with a common lens for discussions across groups. Both 
programmes were whole-school approaches to professional development with 
outside experts and site-based coordinators supporting teachers' practice. 

Many professional development programmes of the past decade engage teachers 
in concrete examination of instructional practice more broadly considered than 
student thinking alone. Chissick (2002) described a three-year professional 
development programme for secondary mathematics teachers in Israel. The goal 
was to change the instructional practices of teachers toward a use of innovative 
instructional practices, including more prominent use of technology and promotion 
of teamwork. Teams from 13 schools were assigned a "facilitator" for one day a 
week for three years, and the school's head of mathematical department worked 
with the facilitator to lead weekly training workshops on innovative instructional 



practices and to assist teachers in experimenting with new instructional practices 
and technology. The facilitators and department heads received support for their 
leadership role in monthly meetings. 

Ghissick (2002) investigated effective implementation of reform mathematics 
teaching practices, effective use of technology and increased teamwork in 
mathematics teaching. The teachers, head of departments, and school heads were 
asked to complete structured and semi-structured questionnaires. Additional data 
collected included weekly reports from project facilitators and field notes of 
observations of meetings. Chissick reported significant change in teamwork culture 
and some changes in classroom practice (specifically the use of more open-ended 
tasks and more student-centred teaching). This study also explored teachers' views 
of themselves as learners. Case studies at two of the schools provided more in- 
depth data analysis and a closer look at a few teachers that included classroom 
observations, interviews regarding teacher beliefs and attitudes toward change, 
mathematics and mathematics teaching, personal history and self esteem. Thus, this 
research included a focus on an individual's role in implementation. 

Another trend of the past ten years has been an effort to take successful models 
of practising teacher professional development and use them with a team of 
practising teachers. Here I focus on successful professional development for 
teachers in East Asia adopted for use in the United States. Teacher development in 
Japan, called kenshu, describes peer collaboration, review and critique of actual 
lessons; this is often referred to as "Lesson Study" (see Yoshida, Volume 1). The 
first thing teachers generally do is establish a lesson study goal. Study lessons are 
collaboratively planned by about four to six teachers, implemented by one teacher 
(usually the teaching is videotaped and observation notes are taken) and afterward 
the lesson is discussed. Administrators and principals participate in this discussion 
because they are considered "peers". The participants discuss their observations 
and decisions about how to improve the lesson. An invited observer provides an 
"outside prospective". An outside observer is most often an instructional 
superintendent, an individual appointed by prefectures to regularly visit schools 
and advise teachers in a region. The outside advisor can also be a university expert 
or teacher on leave hired to provide professional development (Fernandez & 
Yoshida, 2004). In Japan, practising teachers organize the most common type of 
professional development with the formation of a study promotion committee 
drafting a yearly study plan, which is the negotiated with the teachers at different 
grade levels. There are also interschool programmes organized by district-wide 
subject area associations of teachers (Shimahara, 2002). 

Lesson study as practiced in Japan can be obligatory or voluntary. For example, 
one type of obligatory programme is an internship for beginning teachers. 
Internships are legally required for teachers and provide a one-year probation 
wherein beginning teachers have reduced responsibilities and a mentor who is 
selected from among experienced teachers. As part of the internship, interns 
observe mentor teachers teaching and implement study lessons before their more 
experienced colleagues. The beginning teachers learn and practice the different 
roles of the teacher in teaching a lesson (Shimizu, 1999). The internship is 



designed by the prefectural education centre or local board (Shimahara, 2002; 
Yoshida, 2002). Though there are obligatory top-down initiatives, lesson study in 
Japan is embedded in the culture of teaching (Shimahara, 2002). Three premises 
are used as organizing principles for Japanese professional development. The first 
is that teaching is a collaborative, peer-driven process that can be improved 
through this process. The second is that peer planning is critical to teaching. The 
third is that teachers' active participation is a critical element of professional 
development and teaching. Although these premises constitute the normative 
framework of Japanese professional development, school-based teacher 
professional development varies a great deal depending upon the leadership of 
teachers in the local context and the level of teaching (Shimahara, 2002). In 
general, lesson study is utilized more at the elementary school level and is less 
common at the secondary level. 

On the basis of their growing understanding of how lessons are conducted and 
prepared in Japanese classrooms, Hiebert and Stigler (2000) suggested that 
educators in the US consider a form of lesson study to build professional 
knowledge; thus variations of lesson study are rapidly proliferating across the 
country (Chokski & Fernandez, 2004; Fernandez, 2005). As lesson study comes to 
be viewed as effective approach to professional development, school and district 
administrators select this structure as a means to provide a professional 
development experience connected to the classroom work of teachers. Because 
collaboration is not part of the culture of teaching in the US, management and 
teacher educators need to bring teachers together to engage them in lesson study. 
Lessons study requires management to assist with scheduling, allocating funding 
and arranging for substitute teachers and these endeavours are often initially 
obligatory for teams of teachers. 

Podhorsky and Fisher (2007) described a lesson study implementation in an 
elementary school in a low socio-economic area, offered in response to No Child 
Left Behind (NCLB) legislation, a high stakes accountability programme. Lesson 
study was selected because it provides a collaborative environment for teachers to 
focus on curriculum and student learning to facilitate an increase in student 
achievement. Teacher participants (30 teachers of grades 1-5) met weekly for 
roughly a year to plan, teach, and critique lessons. The participants were also part 
of a university class, but the approach became one that was implemented school- 
wide. The school administrator at the time was very involved (Fisher, personal 
communique, June, 2007). 

The teachers were observed as they engaged in lesson study and were 
interviewed individually and in focus groups. Teachers and school site 
administrators were surveyed using Likert-scale questionnaires assessing 
perceptions of this model of professional development. From the surveys and 
interviews we learn about the participants' perspectives on the strengths and 
challenges of the lesson study process within their context. The strengths of lessons 
study as identified by the teachers included: an emphasis on meaningful lessons; an 
impetus for implementing short and long term goals; and a focus on student 
assessment. Teachers also cited the benefits of the community aspect of 



participation in lesson study, increased reflection on teaching practices, and 
excellent preparation for attaining national teaching certification. The challenges of 
implementing lesson study in this urban school were the planning time required for 
the research lesson and, in particular, the significant amount of time required 
outside of the school setting. Lesson study requires common preparation periods 
with grade-level teams, common curriculum, and external guidance in conducting 
lesson study; resources not readily available to teams of practising teachers in other 
schools throughout the United States. However, this situation is slowly changing. 
According to a survey conducted in 2004, a majority of US lesson study groups 
met at least once a week, most during the school day (Chokshi & Fernandez, 
2004). As lesson study becomes part of the culture of teaching in the US, it can 
evolve into a more teacher-led process of professional development. 

In another approach to professional development, teams of practising teachers 
are often brought together to implement challenging curriculum. Balfanz, Maclver, 
and Byrnes (2006) reported on a study of the first 4-years of a mathematics reform 
project, the Talent Development Middle School Model Mathematics (TD) 
programme, which was initiated in three, high-poverty urban US middle schools in 
the context of whole school reform. This programme included professional 
development that was directly linked to implementation of three reform 
mathematics curricula in grades 5 through 8 (10-13 year olds). The mathematics 
reform (curricula, corresponding professional development and whole school 
initiative) was instigated to meet the national milestones of the federal No Child 
Left Behind (NCLB) legislation. The professional development approach was that 
of providing "model lessons" using the curricula. 

The professional development was led by peer teachers and experienced users of 
the curricula and took place over three days of summer training that was followed 
by monthly 3-hour workshops on Saturdays. The monthly sessions were focused 
on lessons that would be taught in the participants' classes the following month. 
The facilitator guided the teachers as participants through the upcoming lesson and 
modelled important aspects, including the mechanics of the activities, questioning 
and then providing an opportunity for the participant teachers to ask questions and 
discuss with each other past experiences of teaching. The teachers also had 
substantial implementation support in the form of curriculum coaches that spent 1- 
2 days a week at a school helping teachers in their classrooms. The curriculum 
coach co-taught, modelled, assisted with lesson planning, provided feedback and 
worked with the teachers to make modifications to the curriculum to address his or 
her students' specific needs. By the completion of the fourth year of the initiative, 
two teacher leaders from each school were ready to help with on-site 
implementation. Part of the teacher-leader training involved shadowing the 
curriculum coaches in their work with teachers. 

Even with all of the support structured into this programme, high levels of 
implementation were difficult to achieve. Across the four years of the study, about 
two-thirds of the teachers achieved the minimum recommended hours of 
professional development each year. The initial goal of having teachers' complete 
instruction in the use of 6 to 8 units of each grade level's instructional materials 



was not achieved. Two-thirds to three-fourths of the classrooms in these high- 
poverty middle schools obtained at least a medium-level of implementation, but 
with significant variation across sites. Of significance to this discussion, the 
Balfanz et al. (2006) noted important factors of institutional context that may have 
affected the implementation: school leadership; scheduling; staffing and resources. 
As an example, during the four years of the reported intervention, only one school 
of the three had the same principal and one school had three principals, with 
varying degrees of commitment to the programme. Staffing patterns changed as the 
administration either successfully or unsuccessfully provided the resources, and 
assigned or reassigned teachers. Teacher turnover was one of the biggest 
challenges; by the fourth year of the study only 31% to 59% of the homerooms 
across the three schools had mathematics taught by a teacher who had participated 
in the reform effort all four years. 


The studies described illustrate themes of mathematics professional development 
for teams of practising teachers in several countries. The programmes are 
structured toward predetermined goals of improving student achievement by 
constructing effective mathematics environments for student learning. 
Consequently, the focus is often on developing understanding of connections 
among concepts and topics in mathematics, increasing awareness of students' 
mathematical thinking and the pathways of development, and supporting teachers' 
changing instructional practice. Whether the approaches to professional 
development involve engaging teachers in doing mathematics, or examining 
student thinking, or introducing lesson study as a mechanism for improving 
teaching, or modelling the teaching of reform curriculum, some common themes 
have emerged. Here we see, teams of practising teachers engaged in programmes 
structured to foster collaboration, including not just peers but heads of departments 
and other significant site leaders. The programmes illustrate a growing awareness 
of the critical role of alignment with organizational context (see Cobb & Smith, 
this volume). The following case study is an example of these common themes and 
importantly, illustrates that the goals and structure of the work are equally affected 
by teachers' and managements' participation. 


The story that emerges from the studies on teams of practising teachers across 
multiple school sites is one of differential enactment for seemingly similar 
programmatic activities, highlighting the co-constructed nature of the professional 
development experience. I have chosen a case study for elaboration. The case study 
describes a programme designed to assist teachers in becoming mathematics 
specialists. The initiative emerged in a context of addressing poor student 
achievement at a local level. The focus was on engaging a team of teachers in 
doing mathematics and supporting reorganization of their instructional practices. 
Several school districts and universities in the United States have partnered to 



develop professional development programmes for mathematics specialists, 
individuals with specialized preparation in mathematics (see e.g., Nickerson & 
Moriarty, 2005; The Journal of Mathematics and Science, 2005). Mathematics 
specialists can have work assignments such as a lead teacher or coach. In some 
specialist work assignments, teachers may teach only mathematics or they may 
teach mathematics and one other subject, such as science. In this way, teachers can 
focus on being knowledgeable or expert in the teaching and learning of one, or at 
most two, subjects. 

Nickerson and Moriarty (2005) described an initiative in which 32 upper- 
elementary teachers in a large urban school district in the United States were hired 
as additional staff at eight low-achieving schools. The teachers, as the only 
teachers of mathematics in these schools, travelled from classroom to classroom 
with carts of materials "visiting" other teachers' classes to teach three mathematics 
classes, each lasting 90 minutes. During their first year of teaching as a 
mathematics specialist, they participated in a professional development programme 
that had a focus on building teachers' knowledge of mathematics and mathematics 
for teaching with connections to practice. The principals arranged for shared 
professional development time at each school site. The 60-90 minutes each day 
was intended to facilitate teacher's sustained growth in knowledge and practice. 
The teachers met for two weeks one summer and then about three hours a week for 
one year in coursework designed to help teachers reconceptualize the mathematics 
they were teaching and to deepen understanding of mathematics pedagogy. The 
coursework also entailed teachers' learning about how students' mathematical 
thinking develops. Coursework and on-site activities provided opportunities for 
reflecting on practice by engaging in collaborative reflective teaching cycles and an 
analysis of student work, their own and from research. In addition, the mathematics 
specialists had on-site coaching support from the teacher educators about once a 

Furthermore, the coursework, site-based support, and daily, shared professional 
development time was intended to facilitate teachers' sustained, generative growth 
in content knowledge and practice. Like other programmes described in this 
chapter, this team of practising teachers was brought together by administrators 
with pre- determined goals. The professional development was in support of a 
larger reform initiative, and was planned with the school district mathematics 
instructional team. The designers of the initiative had the expectation that the 
provision of these resources would promote the formation of teachers' professional 
communities of collaborative, reflective practice. Nickerson and Moriarty (2005) 
used the construct of teachers' professional community as defined by Secada and 
Adajian (1997). Teachers' professional communities are described along four 
dimensions: (I) shared sense of purpose, (2) co-ordinated effort to improve 
students' mathematical learning, (3) collaborative professional learning, and (4) 
collective control over decisions affecting the mathematics programme. The 
researchers undertook an analysis of teachers' activities and experiences as situated 
within institutions as opposed to a structural analysis (Cobb & McClain, 2001). 



Based on an analysis of a number of data sources including weekly field-notes 
of teacher educators and researchers regarding visits to schools, interviews with 
teachers and mathematics administrators, teachers' written coursework and 
reflections, teachers' mathematics exams, a survey regarding curriculum 
implementation, and an interview with the school district leadership team. 
Nickerson and Moriarty (2005) investigated the relationship of professional 
development, encompassing both formal and informal sources of support, to 
teachers' knowledge and practice. Nickerson and Moriarty (2005) reported that 
professional communities formed at some sites and not others. Five aspects of 
teachers' professional lives emerged as significant in the formation and strength of 
teachers' professional community: (1) the relationship the mathematics teachers 
had with the school administration and other classroom teachers, (2) the respect for 
and access to the knowledge of other mathematics teachers, (3) the presence or 
absence of a teacher leader at the site, (4) a shared base of mathematical and 
pedagogical knowledge, and (5) shared high teachers' expectation for students. The 
strength and nature of teachers' professional community is significant in local 
interpretation of reform and the development of collective professional values and 
goals (see e.g., Talbert & Perry, 1994). 

One important aspect of teachers' professional lives was the relationship that a 
team had with a principal and other teachers at the site. The teachers applied for 
positions at particular schools. The school principals made the hiring decisions. At 
a few schools, the teachers were at the site and were encouraged to apply while 
other principals hired all the staff from outside. School-based administrators were 
able to align resources for the teaching of mathematics - choosing teams of 
participants, arranging shared professional development time at each site, and 
facilitating the teams' ability to exercise collective control over decisions 
concerning the mathematics programme. The other teachers further contributed to 
or hindered the formation of a shared vision. At some sites, administrators arranged 
for minimal shared professional development time that was sometimes overtaken 
by other responsibilities. There was evidence of administrators who did not 
understand the goals of the initiative and inhibited the power of teachers to make 
decisions regarding the mathematics programme. 

A second important aspect related to teachers developing respect for and access 
to the knowledge of other mathematics teachers because it involves teachers seeing 
themselves as members of a community with contributions to make to the 
collective capabilities (Nickerson & Moriarty, 2005). This sense of self as a 
valuable member of a team contributed to collaborative professional learning at 
some sites. However, one team of teachers expressed a view that they were 
implementing school district mandates and spent little time together unless 
mandated by an outside presence. A third important aspect was the presence or 
absence of a teacher leader who appeared to play a fundamental role in supporting 
collaborative professional learning and shaping a shared sense of purpose. A fourth 
important aspect was teachers' mathematical knowledge that appeared to affect 
their collective control over decisions regarding the mathematics programme and 
their ability to effectively discuss instruction and student learning. Teacher teams 



differed in how they spent the professional development time together. Some 
teachers used the time to help each other with mathematics and to share the 
successes and failures of their classes. Other teams used the time to look ahead, 
prioritize topics, and examine mathematics across grades. Finally, and 
significantly, shared high expectations for students were successfully jointly 
constructed by some teams and administrators and not others. 

The goals and structure of the work were equally affected by teachers' and 
managements' participation. Teacher educators and administrators designed an 
initiative shaped by administrator's selection of team participants and alignment of 
resources. The programme was jointly constructed by teachers' sense of self as a 
mathematics teacher, teachers' respect for others' contributions, and their ability to 
effectively discuss instruction and student learning. Teacher educators and school 
district administrators designed the initiative, local site administrators decided what 
to require, and the teachers who chose the mathematics specialist programmes 
shaped the initiative by their varying and evolving participation. The case study 
suggests that the fostering of collaboration, the involvement of administrators, and 
the alignment of organizational context do not individually account for differential 
enactment and point to the complexity of studies on teams of practising teachers. 


The focus in this chapter has been on studies of the approaches to professional 
development for teams of teachers, with a particular focus on the past ten years. 
Issues of political, social, and economic significance contribute to the structure of 
mathematics professional development for teams by providing resources for reform 
curricula, new technologies, and orienting administrators' focus on mathematics. 
This survey of studies of the past decade highlights the nature of professional 
development approaches for teams of teachers. The studies illustrate an aim of 
fostering collaboration among colleagues around central issues such as developing 
a common lens for viewing and discussing students' early learning, connections 
among mathematical content areas, and improving instructional practice. The 
studies of the past decade reveal increasing involvement of management, 
principals, heads of department and other stakeholders, as part of crafting the 
vision of effective mathematics teaching. Furthermore, these studies speak to the 
need to align programmes with the organizational context in which teachers work. 

The research that has emerged from the studies point to the complexity and 
highlight the interdependence of institutional context, relevant management, and 
teachers themselves as learners. Teams brought together to foster formation of 
communities of collaborative learners are affected by and affect the different ways 
that teachers participate in the team. Likewise, as Krainer (2001, p. 282) suggested 
the management's vision and alignment of resources critically affects and is 
affected by institutional context: 

Principals and other important stakeholders such as regional subject 
coordinators or superintendents with different roles and functions in school 



system, have their own ideas and beliefs about the nature of learning, 
teaching, mathematical knowledge, and reform. [...] it is essential to pay 
more attention to their role in the professional development of teachers, both 
practically and theoretically. 

The research in social context suggests the importance of consistent intentions 
and motives among teacher educators, administrators, and teachers (Cobb & 
McClain, 2001). Yet, many of these initiatives describe multiple layers of 
administrators, wherein the administrators who decided what to require were 
distinct from the site administrators who are proximate to teachers. As a district 
superintendent describes (Nickerson & Brown, 2008, p. 20): 

We've got. I don't know. We've got 1, 2, 3 partnerships over 26 teachers 
right now. We've challenged principals next year that they all will have at 
least one partnering teacher when school starts in September. Principals are 
uncomfortable with that because they have to figure out how to make that 
happen. We challenge them to make that happen. We think, you know, trust 
us. In the end you will be glad you did. 

Although management selected or required the programmes required, several 
researchers described the varying degrees of commitment by the administrators 
that work most closely with teachers (see e.g., Balfanz et al., 2006; Nickerson & 
Moriarty, 2005; Walshaw & Anthony, 2006). 

In closing, we must acknowledge that the nature of the professional 
development programmes for teams of practising mathematics teachers is reflective 
of different contexts. There are differences in the issues for developed western 
countries with substantial resources directed to large-scale recruitment of teams of 
teachers and developing countries with limited time and resources. Whatever the 
nature of these professional development programmes, as Perrin-Glorian, DeBlois, 
and Robert (this volume) conclude, it is important to study teaching within the 
larger societal and political context and to recognize that when practising teachers 
must change because of externa) reforms, it can be very difficult to integrate new 
instructional practices with institutional and social expectations. This suggests that 
mathematics educators and researchers can learn much by taking up the challenge 
of understanding the complex interrelationships among teachers, peers, teacher 
educators, administrators, and institutional contexts. 


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Susan D. Nickerson 

Department of Mathematics and Statistics 

San Diego State University 





Studies on Their Professional Growth 

Focusing on participants in mathematics teacher education, this chapter focuses 
on face-to-face learning communities of prospective teachers. Important elements 
of a learning community synthesized from the literature are addressed in the start- 
ing section and used as a frame for discussion in the final section. Two cases from 
Taiwan and Portugal, which comprise prospective teachers for secondary and 
elementary schools, are presented. Examples of learning communities both initi- 
ated by participants and created by institutional arrangements are given, with con- 
sideration for their outer contexts. The process and product of learning from one 
another within a community from the two cases are synthesized with respect to the 
inner issues of face-to-face learning communities. 


The responsibility of mathematics teachers is to foster students' learning. However, 
teachers and prospective teachers are also learners. In particular, prospective teach- 
ers have to learn how to carry out their job and later, as practising teachers, need to 
learn to deal with new mathematics topics, technologies, needs from students, and 
demands from curriculum and society. Both teachers and prospective teachers learn 
- in practice - for practice, and - from practice. For prospective teachers, this in- 
cludes not only teaching practice in school contexts, but also learning practice dur- 
ing university courses and in informal situations. They learn as they carry out the 
activities set up in the courses of their teacher education programme and during 
field work as they design instructional units and educational materials and tasks, 
observe classroom situations, interview students and work together with their col- 
leagues and supervisors. 

In this chapter, learning is viewed as both an individual and a social process. 
People learn as they interact with the physical and social world and as they reflect 
on what they do. Therefore, learning originates from the activity of an individual 
carried out in a given social context (Lerman, 2001). Prospective teachers learn 
from their activity and their reflection on their activity, and such learning takes 
place in a variety of places, as they interact with others, notably their university 

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 111-129. 
© 2008 Sense Publishers. All rights reserved. 


teachers, colleagues, school mentors, school students, and other members of the 

A {earning community is a special context in which practising teachers and pro- 
spective teachers may learn. The most important feature of a learning community is 
that its members learn from one another. There may be differences in age, experi- 
ence, status, and professional roles, but the unifying element is that all assume that 
they are learners; they are keen to learn together and, most importantly, to learn 
from the others. 

These learning communities may take a variety of forms. They may develop 
from a group that developed habits of working together or a group that was particu- 
larly constituted for the purpose of learning or carrying out some project. Such a 
group may be formed spontaneously by the initiative of its participants, or may be 
created by some institutional arrangement. Therefore, the group of mathematics 
teachers in a school, formed on a purely administrative basis, may become a learn- 
ing community if the teachers begin valuing the fact that the participants can learn 
a great deal from each other. Similarly, a class for prospective teachers, that by 
itself is a highly contrived setting, can constitute a learning community if the in- 
structor and the prospective teachers develop relationships of learning from each 
other (Jaworski, 2004). 

People may work together in a variety of ways, either competitively or collabo- 
ratively. For example, they may constitute teams, that "are mostly selected by the 
management, have pre-determined goals and therefore rather tight and formal con- 
nections within the team" (Krainer, 2003, p. 95); In contrast, communities "are 
regarded as self-selecting, their members negotiating goals and tasks. People par- 
ticipate because they personally identify with the topic" (p. 95). A group that be- 
gins as a team may develop a working culture that transforms it into a community. 
Learning is generally regarded as a key element in becoming a member of a com- 
munity but it may also be regarded as an important feature of the activity of the 
whole community. Therefore, a learning community is a community where learn- 
ing is valued as an important outcome of the groups' activity. 

The notion of learning community in many respects is close to that of discourse 
community. For example, for Putnam and Borko (1997), a discourse community is 
a group of people that learned to think, talk, and act in a similar way - in our case, 
as mathematics teachers. The authors suggest that prospective teachers can model 
joint cognitive activities, doing "careful analysis of the cognitive roles played by 
each participant" (p. 1276). Another related notion is that of community of practice 
(Lave & Wenger, 1991), where newcomers learn from old-timers by participating 
in the tasks that relate to the practice of the community and, with time, newcomers 
move from peripheral to full participation. Still another related notion is that of 
inquiry community (Jaworski, 2005), in which a group of professionals question 
their existing practices and explore alternative practices. In this chapter, we use the 
notion of learning community because our focus is on prospective teachers' face- 
to-face learning in formal and informal teacher education activities. 

Diversity and heterogeneity among the participants of a learning community of- 
ten make it difficult to find a common language, to adjust purposes and ways of 



working. However, such diversity may be very beneficial to the work of the com- 
munity. As Ponte et al. (in press) highlight, different experiences, viewpoints, and 
expertise may make the group more powerful in identifying and generating solu- 
tions to deal with relevant issues, thus leading to quite significant learning from the 
participants. For example, Clark and Borko (2004) describe how a community was 
generated in a professional development institute for middle-school mathematics 
teachers that focused on algebra content knowledge. In their view, the participants 
at the beginning formed a rather diverse group but the tasks proposed on the first 
day of the institute fostered their active participation in activities such as "explain- 
ing and clarifying ideas, building off of others' ideas, admitting weaknesses, giving 
praise to others, and laughing" (p. 229), that the authors regard as indicators of the 
establishment of a community. 

Working with colleagues, aiming at professional learning, is also an essential 
part of the model proposed by Hiebert, Morris, and Glass (2003, pp. 21 1-212) for 
generating and accumulating knowledge for teaching and for teacher education 
(and we may see prospective teachers and practising teachers both involved in this 

Becoming a professional teacher, in our view, means drawing from, and con- 
tributing to, a shared knowledge base for teaching. It means shifting the focus 
from improving as a teacher to improving teaching. This requires moving 
outside the individual classroom, surmounting the insularity of the usual 
school environment, and working with colleagues with the intent of improv- 
ing the professional standard for daily practice. This also requires redirecting 
attention from the teacher to the methods of teaching. It is not the personality 
or style of the teacher that is being examined but rather the elements of class- 
room practice. 

Jaworski and Gellert (2003) indicate that, to teach effectively, prospective 
teachers need to integrate different kinds and layers of knowledge. Such knowl- 
edge develops through the work of the university and school, of prospective teach- 
ers and their university tutors, mentors and other teachers. In their view, all the 
participants involved, including prospective teachers, may contribute to this devel- 
opment: "all participants are learners, their roles developing in relation to their 
critical evaluation of them" (p. 270). 

Ponte et al. (in press) indicate that there arc four key issues in learning commu- 
nities. The first is the purpose of the group and its relation to the personal purpose 
of its members. In fact, a very important condition for one person to learn is his or 
her wanting to learn and, similarly, a very important condition for one group to 
learn is its desire to learn. Therefore, we need to consider the purposes that led to 
the creation of the group, to what brought people to the group and how they iden- 
tify with the purposes of the group. And, of course, we need to consider the fact 
that the group's purposes change with time. The second issue concerns the knowl- 
edge that develops from the activity of the community based on shared practices. 
We need to pay attention to what participants are really learning and how they 



learn it. Is it superficial or significant knowledge? The third issue is how learning 
happens in the group. Do group activities involve intensive moments of studying, 
discussing, and reflecting, or are the participants mostly carrying out routine activi- 
ties? Is learning developed through negotiating meanings and sharing reflections or 
just by memorizing and imitating the others? Finally, the fourth issue concerns the 
roles and relationships of the participants. In a learning community, the mutual 
involvement and commitment of members to the progress of the group are essential 
features (Ponte et al., in press): 

The learning community is stifled if some members do not feel confident 
enough to expose their concerns, do not ask for help, and refrain from par- 
ticipation in the group or if, on the contrary, other members participate "too 
much", occupying all space, helping others too much or in an improper way, 
etc. A proper style of leadership is a critical element to the working of any 
group [...]. There are always participants that play a more prominent role in 
one stage or another of any group, but the group itself may establish a collec- 
tive leadership, assuming the most important decisions after a thorough dis- 
cussion of the issues, and a distributed leadership for practical activities, as- 
signing specific group members the conduction of such or such activity. 

The development of e-learning and other distance education settings generated a 
great deal of interest in learning communities for practising and prospective teach- 
ers (Llinares & Vails, 2007; Ponte, Oliveira, Varandas, Oliveira, & Fonseca, 2007; 
Borba & Gadanidis, this volume). However, much less attention has been paid to 
face-to-face learning communities of prospective teachers; this is the focus in this 
chapter (see also Llinares & Oliveiro, this volume). We begin by presenting two 
case studies that illustrate learning communities for prospective teachers as learn- 
ers. The first, from Taiwan shows how prospective secondary and elementary 
teachers form learning communities to study for exams, and the second one, from 
Portugal, illustrates the different learning communities to which a prospective 
teacher may belong during their teacher education programme. 


In this section, the social and political contexts of prospective teacher education in 
Taiwan are described. Based on the specific conditions under which prospective 
teachers are certified, certain learning communities are formed. 

The Supply of Prospective Teachers in Taiwan Exceeds the Demand 

According to the statistical data of 2006 annual reports on teacher education (Dai, 
Kuo, Yang, Lin, & Wei, 2006), 72 universities and colleges with teacher education 
programmes, in general, a two years study, graduated a total of 17,000 certified 
elementary and secondary school teachers. Among them, about 4,000 were selected 
by schools as regular teachers. From the 13,000 of prospective teachers who were 
not selected, about 2,500 are teaching in schools as substitute teachers and more 



than 10,000 do not have any teaching position in schools. These teachers are 
named "stray teachers" by the media for the fact that they are certified as teachers 
but are not hired by any school. This has become a social issue. 

The social status of teachers in Taiwan is relatively moderate; teachers are gen- 
erally deemed as middle class. Their initial salary is about 30% more than that in 
almost all other jobs; those who specialize in humanities may even exceed other 
humanity-related jobs by 80%. Teaching is generally a lifelong career. With a sta- 
ble pay, few change for another career as the Taiwanese society has an unstable 
economy. A reliable job with a good retirement system is attractive to young peo- 
ple. A teacher qualified to retire will have a monthly pension that is 85% to 95% of 
his or her original pay. Therefore, it is likely that a teacher with 30 years seniority 
would receive retirement pension for another 30 years. These incentives attract 
many distinguished youngsters to teacher education programmes. Besides, low 
birth-rate and concomitant low population growth rate in Taiwan bring about a 
decrease every year in the number of people in each age group, from the maximum 
of about 420,000 births in 1976, to a minimum of 205,000 births in 2006. Due to 
this, the number of school-age children lessens each year, and so does the demand 
for teachers. While the demand for teachers declines, the incentives for people to 
become teachers remain strong. Under such circumstances, when the number of 
certified prospective teachers that graduate is not efficiently limited, the supply of 
teachers naturally exceeds the demand. 

Examination for Teacher Certification 

Owing to the new social issue of "stray teachers", the Ministry of Education set a 
limit on enrolling students for teacher education programmes. At the same time, 
the Ministry of Education started to evaluate these programmes. The ones that 
score low would have their enrolment decreased at first. Two years later, they 
would be re-evaluated. The ones that fail to pass the evaluation would have to close 
their education programmes. In addition, the Ministry of Education instigated an 
examination for teacher certification. Before 2005, prospective teachers became 
certified teachers once they completed the teacher education programme which 
included one year of teaching practice in school. Since 2005, however, the period 
of teaching practice for prospective teachers was cut in half, and the examination 
for teacher certification was added and administered by the Ministry of Education. 
Those who pass the exam receive the teacher's certificate. 

The examination for teacher certification comprises of four subjects: Mandarin- 
Chinese; general education theory and system; youth (child) development and 
counselling; and secondary (elementary) curricula and pedagogy. Words inside and 
outside of parentheses are the subject names for elementary school teachers and 
secondary school teachers respectively. In 2006, there were 7,857 persons regis- 
tered for taking the exam; 4,595 of them passed it, which means a passing rate of 
58% (Dai et al., 2006). 



Examination for Teacher Selection 

Following the examination for teacher certification, those who get the teacher certi- 
fication attend the examination for teacher selection held in each of the 23 counties 
and cities. The examinations for senior high school teacher selection are hosted by 
each school. There are two phases to the selection exam. The first phase tests the 
prospective teachers' content knowledge (e.g., mathematics teachers will have an 
exam on mathematics). Those who pass the first phase can proceed to the second 
phase, which is a teaching demonstration and interviews. Generally speaking, on 
average, each elementary level prospective teacher takes more than three examina- 
tions for teacher selection. The number of those that obtain a teaching position is 
between 1% and 2% of the total. Secondary school teachers are a little better off. 
The ratio of those that obtained a new teaching position is around 5% to 6% in 

Du Shu Hui (BltW), Learning Communities for Exams 

In the following, we regard the prospective teacher education programmes at Na- 
tional Taiwan Normal University. In mathematics education, the university pre- 
pares for the examinations for teacher certification and teacher selection as well as 
for the entrance examination for graduate schools. Personal communication with 
colleagues and prospective teachers about how the graduates prepare for the ex- 
aminations mentioned above, indicated that active students usually form Du Shu 

Hui (IKftft), or learning communities. Prospective teachers participating in Du 
Shu Hui are more likely than others to pass these examinations. Du Shu Hui are 
usually initiated by one or several students. The motive of initiating a Du Shu Hui 
is to prepare for certain examinations. Many of these Du Shu Hui last nearly a 
whole year. Under the present social and political contexts of Taiwan, these pro- 
spective teachers, in order to further their education or to become teachers, form 
their Du Shu Hui as a learning strategy. The interactions and the norms for interact- 
ing, the knowledge sharing, the operation of collaborative learning, and the out- 
come and feedback of the learning communities are interesting issues in terms of 
social dimension. 

Du Shu Hui is a Chinese noun (9ltt0) meaning study group. If translated liter- 
ally, Du means to read, Shu means books and Hui means meeting. According to the 
definition of learning community given by Krainer (2003), Du Shu Hui is a learn- 
ing community in that: 

- Du Shu Hui are self-selecting, their members negotiate goals and tasks. 

- Prospective teachers participate because they personally identify with the 

The following is an overview of some Du Shu Hui we picked out as examples of 
how these work as learning communities. 

Method of collecting data. In order to understand prospective teachers' Du Shu 
Hui, a structured interview was conducted in November, 2007. The structure of the 



interviews consists of three parts: the incentives and goals for setting up learning 
communities, their organizational operation, and the outcomes and reflection of 
these communities. 

Subjects from departments of mathematics (education) in three universities, two 
for elementary school teachers and one for secondary school teachers were selected 
and interviewed. One of the mathematics educators from the departments of these 
subjects conducted the interviews. Each interview lasted about 30 to 40 minutes. 

Four Examples ofDu Shu Hui. After being certificated, most of prospective secon- 
dary mathematics teachers take examinations either for teacher selection or for 
further study in graduate institutes, majoring in mathematics or mathematics educa- 
tion. In the examination for secondary mathematics teacher selection, solving 
pedagogical mathematics problems, designing instruction units and 15-minute 
teaching demonstration are the main activities. From the data we collected, we 
chose three Du Shu Hui, each of which respectively focused on one of the three 
activities during their community meetings. A brief description of the Du Shu Hui 
are shown in Table 1. This table also shows one Du Shu Hui which focused on 
studying advanced mathematics for entrance examination of graduate institute. 

Some specific features of each Du Shu Hui are described as following. During 
each community meeting, the five members of the Du Shu Hui teaching demon- 
stration took turns to demonstrate 15 minutes teaching. The other four members 
could spend as much time as they wanted on commenting others' teaching demon- 
stration. They are critical friends (Jaworski, 1999) to each other. During the inter- 
view with Shi-An, he reflected that "I was influenced by other members. They 
helped me rectify my teaching approach. This is rather important. I wouldn't have 
passed the exam if not for the rectification." and "everyone has blind spots. The 
members helped me see mine." 

In Du Shu Hui on designing instruction units, the reason there were only two 
members was that each one would like to design more instruction units to under- 
stand more fully the teaching content. One of them has to travel more than 100 
kilometres to meet the other at a cafe each week. Both of them were selected as 
senior high school mathematics teachers in the summer of 2007. Though they are 
teaching in different schools, they remain close friends. One member reflected in 
the interview that: "Now, whenever I have problems concerning teaching I talk it 
over with Wen-Rong. We discuss about issues including mathematical contents, 
teaching method, and developing test items." 

In the Du Shu Hui for solving pedagogical mathematics problems, four mem- 
bers are still in their fifth practicum year. They meet one whole day a week. They 
are solving pedagogical mathematics problems appeared in previous examinations 
for teacher selection. Because one of them practice teaching at the best gifted class 
in Taiwan, tough problems very often are assigned to those gifted students to do; 
afterwards they discuss students' answers and learn from those students. The 
community implicitly consists of not only the four prospective teachers but also the 
class of more than 40 mathematically gifted senior high school students. Those 
gifted students are also learning by solving problems. The members we inter- 



viewed expressed that "One of us has very good understanding of mathematical 
concepts. His clear linking of geometry and algebra made a great impact on me." 
and "One of us has excellent conceptual generic examples; they are inspiring." 

Table 1. Examples ofDu Shu Hui, prospective secondary teachers 


Du Shu Hui for exams 

Teaching dem- 

Designing in- 
struction units 


Solving peda- 
gogical mathe- 
matics problems 


To pass teacher 

To pass teacher 

To join 
graduate study 

To pass 
teacher selection 







September 2006 
-June 2007 

December 2006 
-March 2007 

November 2006 
-March 2007 

March 2007~on 


1 weekly meet- 
ing (3-4 hours 

Saturday mor- 
ning biweekly 
(3-4 hours 

2 meetings 
weekly (3-4 
hours each) 

A whole day per 


Took turns to 
practice teach- 
ing (15 
min/person) + 
commented on 
others' demon- 

Discussed over 

their pre- 
written instruc- 
tion units 

- divided the 
textbook content 
by schedule 

- took turns to 

- collected hard 
questions and 
solved by a par- 
ticular member 

- focused on 
doing test ques- 
tions before the 

- do past exam 
questions (2-3 
exam papers) dur- 
ing each meeting 

- (20 min. demon- 
comment) * 4 

- discuss the tough 
questions assigned 
to the gifted stu- 


All passed 

All passed 

3 joined graduate 

study; 1 became 

a teacher 

will take exams in 

In the Du Shu Hui for advanced mathematics, the four members started to take 
turns to give lectures. But gradually most of tough problems in the textbook were 
collected to be solved by one of the members. He became a preceptor for the com- 
munity. He himself expressed that: "It is only by explaining the texts to others in 
detail does one know where his/her weaknesses lie. Other people's criticisms and 



opinions help one to find misunderstandings of texts. This is the best part during 
the discussions." 

Three of them passed the entrance exam for mathematics graduate institute, the 
other one passed the exam for teacher selection. Two members were interviewed 
and expressed that: "By attending this learning community, the members have 
more incentives to study because they oversee each others' progress - one would 
be urged to study when seeing others do so." The advantage of keeping the com- 
munity small is: "They could hear their peers share any mathematical thoughts. 
They were impressed by some of the brilliant thoughts." 

Examples of Learning Communities of Prospective Elementary Teachers 

After interviewing prospective mathematics teachers from two education colleges 
that educate elementary school teachers, it is found that learning communities in 
these schools are not as common as in NTNU. It is worth investigating whether 
these students are less keen on taking the exams due to low passing rates (1% to 
2%) or the university traditions (10 years ago prospective teachers from these col- 
leges were assigned teaching positions after they graduated). However, 30 of the 
graduates (about 5% of the total graduates) from National Taipei University of 
Education, one of the two universities interviewed, passed the examination for 
elementary teacher selection of year 2007. There were about 300 vacant positions 
in 2007. NTUE had a celebration for the graduates' "good performance". Some of 
the students of NTUE said that they joined a school club named "Math Camp". The 
objective of Math Camp is that during summer and winter vacations the partici- 
pants go to elementary schools in remote areas to provide social service for the 
students there. The organization and operation of Math Camp basically follow 
Krainer's definition of learning communities (2003). The following are brief re- 
ports of the interviews. 

In Table 2 is a brief description of two learning communities of prospective ele- 
mentary teachers: Math Camp and Du Shu Hui for enhancing understanding 
mathematics. We will further describe some specific features of those two commu- 
nities as follows. 

In the Math Camp community, prospective elementary teachers are practising 
school organizational operation. Members are put into groups of five to ten and 
organized in structure similar to a school administrative system. Particularly, each 
member is designing mathematics activities. During community meetings, they 
discuss over each member's design of activities. Those activities intend to imple- 
ment to an elementary school for the children there during summer and winter vo- 
cation. The schools chosen for running a math camp are in remote districts. A 
group of them reflected that 

By trying to design various mathematical learning activities, we acquire 
knowledge that is not in the elementary mathematics teacher education pro- 
gram. Since the camp is held in different places every year, we learn about 
regional differences. For example, we have seen the cultural diversities in the 



students of Hakka village schools and coastal Min-Nan schools. These diver- 
gences contribute to the differences in students' response in learning. This 
helps us know that it is necessary and important to teach students in accor- 
dance with their aptitude. 

Table 2. Examples ofDu Shu Hui, prospective elementary teachers 


Du Shu Hui 

Math camp 

Enhancing mathematics under- 


To learn mathematics teaching, ob- 
tain collaboration experiences, and 
enhance member-to-member rela- 

To enhance undergraduate 
mathematics understanding 


5- 10 per group 
(to recruit new members yearly) 

( 1 sophomores + 1 junior) 


All the year round 

September 2007- 
(during school terms) 


Fixed time, once a week 

3 hour-meeting, once a week 


- hold mathematics camps for ele- 
mentary school children during 
summer and winter vacation . 

- activity design, activity execution, 
and novice member training 

- the junior takes position as the 
instructor; the sophomores dis- 
cuss over his or her instruction 


- members relationship enhanced 

- knowledge in the non-elementary 
mathematics curriculum acquired 

- cultural diversities learned 

- the junior: the only one to pass 
the first quiz 

- concentration on studying raised 

The Du Shu Hui for enhancing undergraduate mathematics understanding, ten 
sophomores and one junior mathematics education students have participated. The 
community meeting have developed in the way that the junior student takes posi- 
tion as the instructor which others discuss over what he or she speaks. It seems that 
an instructorship is often developing whenever a community is focused on ad- 
vanced mathematics. The junior student responded to the interviewer that 

I have to go through the texts and questions in advance and then teach the 
others during each meeting. While I teach 1 examine whether I was on the 
right track. I have gained more than others during the process (teaching be- 
comes learning). Among the members 1 was the only one to pass the first 



Furthermore, he expected that "Hope that the sophomores would go over the 
questions before each meeting, otherwise they would not have much progress." 


Teacher Education in Portugal 

The political and social situation of Portugal bears some resemblances but also 
important differences to that of Taiwan. Like Taiwan, in Portugal the population is 
aging and declining and every year the need for teachers decreases, both at the 
primary and secondary school levels. Recently, there was a national effort to uni- 
versalize education and thus new institutions were created to supply the necessary 
teachers; there was a shortage in many school subjects, including mathematics. 
Teaching was then became an attractive career, as it not only provided a reasonable 
salary but also had unique features of flexibility, reduced weekly schedule and ex- 
tra holidays in Christmas, Easter and Summer. All this changed dramatically in the 
last few years. The shortage is over and now there is a large surplus, with many 
unemployed teachers that can do not find a place in schools. The weekly schedule 
was extended and teachers are now required to participate in school activities even 
when students are absent. The government has announced an external mathematics 
exam to ascertain if the teachers to be recruited have the necessary mathematical 
competence. The Government also decided to adopt the Bologna framework, 1 and 
revised the structure of teacher education. Programmes to prepare school teachers 
take five years (for secondary school candidates) or four and a half years (for pri- 
mary) to complete and teachers now get a professional masters' degree. Profes- 
sional practice (or practicum) in schools was reshaped to consist of about three- 
fourths of a year from one full year) but prospective teachers now have to write an 
extended report to conclude their study. 

These changes are taking place at this moment in time; thus it is too early to 
know what the effects will be. So far, the main noticeable effect is a decrease in the 
number of teacher candidates, especially at the secondary school level - due to the 
large number of unemployed teachers and also the long period of time that it took 
the government to certify the new study plans that were submitted by all teacher 
education institutions. 

The Teacher Education Programme of the Faculty of Sciences, University of 


prospective mathematics teacher education programme of the Faculty of Sci- 
s of the University of Lisbon prepares for teaching at grades 7 to 12. This five- 

The Bologna framework is a movement of general reform in higher education in Europe, also includ- 
ing some non-European countries, aiming to promote student mobility, programme comparability and 
renewal of teaching and learning processes. 



year programme, as it happens in other countries, has a three-stage model: (i) Dur- 
ing the first three years, prospective teachers follow scientific-oriented courses 
(covering the standard branches of pure and applied mathematics, with emphasis in 
advanced algebra and infinitesimal analysis); (ii) In the fourth year they take edu- 
cational courses, some addressing general educational issues (pedagogy, psychol- 
ogy, sociology, history, and philosophy of education) and some others dealing with 
mathematics education issues (mathematics curriculum, instructional materials, 
classroom work, assessment, and teaching number, algebra, geometry, statistics 
and probability); and (iii) The most important part of the fifth year is a supervised 
practicum in a school. During the first three years, with only a few exceptions, pro- 
spective teachers take the same courses as pure mathematics majors. During the 
fourth year, the programme seeks to provide prospective teachers with theoretical 
frameworks to analyse educational issues with special attention to current prob- 
lems, and to provide the essential elements to plan and carry out the daily activity 
of a mathematics teacher. It also puts prospective teachers in contact with educa- 
tional practice through two fieldwork courses (one in each semester). The fifth 
year, the programme includes a year long practicum, during which prospective 
teachers are responsible for teaching in one class and become progressively in- 
volved in all aspects of the professional activity of a mathematics teacher. The pro- 
gramme still includes other elements that complete the educational, scientific, cul- 
tural, and ethic preparation of prospective mathematics teachers. Elements for the 
following examples were drawn from different studies carried out by the second 
author, as part of a research programme in teacher education. 

Face-to-Face Learning Communities in Regular Courses 

Within this programme, prospective teachers have several opportunities to consti- 
tute face-to-face learning communities. Each course, during the first four years, 
either in mathematics or in educational subjects, has made such possibility implicit. 
Often, prospective teachers create informal groups to study, to work on assign- 
ments, or to carry out more extended tasks. In the fourth year, most education 
courses explicitly encourage these groups and some of them constitute rather stable 
communities of prospective teachers that tackle in turn the tasks related to different 
courses. One of the most interesting opportunities for prospective teachers to con- 
stitute learning communities is provided in the fourth year by the fieldwork course 
Pedagogical Actions of Observation and Analysis. 2 In each semester, the course 
runs for 3 hours a week, providing an opportunity for prospective teachers to re- 
flect about educational phenomena based on school observations. The aim is that 
they begin to regard these phenomena from the point of view of the teacher and to 
develop their capacity of observation and reflection about educational situations. 
Contrary to all other disciplines of the programme, this one does not have a fixed 

In Portuguese, AccSes Pedagogicas de Observac3o e Analise (APOA). 



curriculum. Its activity is mostly based on observing and reflecting about observa- 
tions and is jointly undertaken by prospective teachers and instructors. Given the 
nature of the work, the classes have between 12 and 16 prospective teachers. Group 
work - involving usually three or four prospective teachers - is the most common 
working pattern used all through the year. Visits to the school are first prepared, 
then carried out, and later discussed in classes at the university. Prospective teach- 
ers present in class the results of extended observations on issues of their choice. 
The most usual form of classroom interaction is informal discussion with active 
participation by prospective teachers. The role of the instructor is to propose tasks 
and to lead discussions. Each semester ends with presentation and discussion of 
projects (in oral and written form). 

The group of 12 to 16 prospective teachers, together with the instructor, consti- 
tutes a learning community. Several factors contribute to that. The fact that there is 
no prescribed curriculum enables that the planning of work be carried out in a 
flexible way with the contribution of all participants. Since the activity extends for 
a full academic year, there is plenty of time for the participants to get to know each 
other and to adjust to the working requirements of this course. Most of the prospec- 
tive teachers experienced working in small groups from the time when they were in 
high school. They now come back to this kind of activity for which most of them 
adjust rather quickly. The balance between the moments of working as a whole 
group and in small groups of 3-4 elements, all with their own more specific divi- 
sion of labour, has proved adequate for carrying out tasks such as observing, re- 
cording data, analysing observations, and reflecting. 

A study by Ponte and Brunheira (2001) indicates how some prospective teachers 
regard this activity: 

While I circulated in the corridors, among pupils, teachers, and staff, I had the 

opportunity to look at things differently and see things that I had never noticed 

before. (Beatriz) 

This visit [to the school] [...] now made me enter a world that I already knew, 

but with other eyes, in another role, a little [as I will do] in the future as a 

teacher. I no longer felt like a pupil although I [still] do not feel like a teacher. 


The observation allowed us to look at the classroom in a completely different 

way, a "teacher's" look. It was there that we began paying more attention to the 

type of class, to the physical conditions, to the teacher's methodology, to the 

pupils' reactions. (Eduardo) 

Although I left secondary school 5 years ago I can already see that it went 

through great changes [...]. (Dora) 

The practicum in the fifth year constitutes another important event in this pro- 
gramme. The prospective teacher, together with one, two, or three other colleagues 
is assigned to a school. He or she is responsible for teaching two classes, and at the 
same time participates in seminars and other activities with his/her supervisors 
from the school and the university. In each school, a micro learning community is 



formed by this small group of prospective teachers and the school supervisor. The 
university supervisor is not present on a daily basis but tends to become more of an 
external consultant. The practicum plays an essential role in developing the profes- 
sional competencies of the teacher candidate and in supporting the construction of 
his or her professional identity, and promoting a reflective and active professional 

The different practicum groups, together with their supervisors, meet regularly 
about once a month. This large group (its size may vary from 30 to 50 or even 70 
participants) plans the programme of work for the year, discusses issues that 
emerge from the activity of the practicum groups, and invites outside experts 
(sometimes a secondary school mathematics teacher) to carry out seminars on spe- 
cific topics. The most important activity is the meeting organized for the end of the 
year, with a format similar to that of a mathematics teachers' professional meeting, 
in which prospective teachers present to each other some aspects of their work, 
through posters, oral communications, and workshops. 

The practicum group constitutes another kind of learning community, especially 
when there are collaborative relationships between the prospective teachers and the 
supervisors that provide the appropriate challenge and support. For many prospec- 
tive teachers, the practicum is the most important element in their teacher educa- 
tion programme; this is understandable, since it provides a confrontation with the 
reality of practice, requiring the mobilization, revision, and integration of previous 
knowledge developed in separate experiences and leading to the development of 
new practical knowledge necessary to conduct the professional activity. One pro- 
spective teacher, Nelia (the study is described in Ponte et al., 2007), indicates that 
one of the aspects that contributed to the success of her practicum was the fact that 
she already worked with her two colleagues in many university courses. She also 
indicates that at her school there were two other mathematics practicum groups and 
there was a good collaboration among all of them. 

An Informal Face-to-Face Learning Community Setting 

Nelia reports that, as a prospective teacher in the last two years of her studies, she 
participated in a research project in mathematics education. This project, conducted 
by a university mathematics teacher educator, involved practising and prospective 
teachers as well as a large group of doctoral students. Overall, the project had 
about 40 members organized in different sub teams that provided intensive activity 
of planning research studies, collecting data, presenting seminars, writing and dis- 
cussing papers; thus this was another important learning community in Nelia's pro- 
fessional journey: 

In this project we have several themes, we analyse several things. Besides 
maintaining a contact with on-going research [...] it is an opportunity to talk 
to people with rather different backgrounds. Therefore, we have a meeting 
once a month and in those meetings we may share the experiences concern- 
ing classroom practice as well as concerning research. Therefore we may al- 



ways talk to somebody, share our questions with somebody, know what the 
others are doing, what questions they also have, and this keeps my inquiry at- 
titude active, that leads me searching for more things [for my practice]. 

It is interesting to note how this project had a profound impact on this prospec- 
tive teacher. The formal activities of the project were important, as were the infor- 
mal contacts with other members and especially the work that she carried out in 
collaboration with her close colleagues. 

In summary, this mathematics teacher education programme provides several 
opportunities for prospective teachers to participate in different face-to-face learn- 
ing communities. These communities vary in size, from very small to rather large 
groups, vary in the intensity of their activity, and also vary in the extent to which 
they support prospective teachers' learning. The most successful learning commu- 
nities seem to be those that combine some sort of formal aims and structure with a 
significant flexibility in carrying out the activities and involve different levels of 
working together from small groups for undertaking specific tasks to large groups 
for sharing and discussing more general issues. 


In order to make sense of learning communities for prospective mathematics teach- 
ers, the outer contexts and inner issues need to be considered. In the following, we 
focus on these two aspects separately. 

The Outer Contexts of Face-to-Face Learning Communities 

A learning community often is rooted meaningfully in its outer context. On the one 
hand, different functions of communities may be established with respect to differ- 
ent relations of supply and demand of beginning mathematics teachers within a 
society. Popularly established Du Shu Hui with the goal of preparing for examina- 
tions in the case from Taiwan resulted from the outer context of a supply of teach- 
ers far greater than available teaching positions. However, whenever the supply is 
less than the demand, establishment of a face-to- face learning community often is 
created by institutional arrangement akin to a team and then transformed into a 
community. The case from Portugal provided an example of learning communities 
that were organized along such approach. An educational system in which an over- 
supply of teachers exists, prospective teacher education programmes are naturally 
extended to include the need for students to perform well on examinations for 
teacher certification and teacher selection. Various types of learning activities may 
take place from each component in such a programme: participating within a learn- 
ing community is one such activity. Examples of learning communities taking 
place in regular courses, research projects and teaching practices are shown to be 
major activities as in the case from Portugal. Thus, the two cases, Taiwan and Por- 
tugal, provide a diversified set of examples of learning communities with regard to 



the outer contexts of the different relations of the societal need for teachers, and the 
components in a broader sense of prospective teacher education programme. 

The inner Issues of Face-to-Face Learning Communities 

Four inner issues for learning communities were identified by Ponte et al. (in 
press) at the beginning of this chapter. However, a more fundamental principle is 
that, in both the cases from Taiwan and Portugal, apart from each of distinct goals 
for learning, were similar learning communities in the sense that the prospective 
teacher participants were learners that learned from one another. The following 
provides a synthesis from the examples of the learning communities given by the 
two cases mentioned according to the four inner issues. 

The first issue is the purpose of the group and its relation to the personal pur- 
pose of its members. The cases of Taiwan and Portugal both illustrate that the aim 
of the learning communities is for the professional learning of prospective teachers 
by generating and accumulating knowledge about teaching and teacher education, 
as is indicated by the model proposed by Hiebert et al. (2003). The members of the 
learning communities in these cases did participate in framing their tasks. This is 
clear in the case of Taiwan. In the Portuguese case, prospective teachers from the 
field-based course Pedagogical Actions of Observation and Analysis learned 
through the process of observing and reflecting on educational situations, as well 
during their practicum activities and in the research project. In all cases the tasks 
were negotiated by participants with teacher educators or project leaders. 
The second issue concerns the knowledge that develops from the activity of the 
learning community based on shared practices. Learning activities within a learn- 
ing community can be sequentially separated into three phases: entry, interacting 
and reflecting. Professional knowledge of teaching can be distinguished into prac- 
tice knowledge and thought knowledge. Practice knowledge is context-dependent 
and has to be gained from real teaching practice (Goffree & Oonk, 2001). Thought 
knowledge is revealed when one is thinking about teaching without facing stu- 
dents. At the entry phase of learning communities provided by the Portuguese 
field work course, participants focused on practice knowledge. In Taiwan, at the 
entry phase of Du Shu Hui for examinations, participants designed instruction units 
and demonstrated teaching which focused on thought knowledge. To what extent 
that different entry knowledge might influence the knowledge that is generated and 
accumulates during interacting and reflecting phases is worth further investigation. 
For example, at the entry phase, participants in the Math Camp designed various 
mathematical learning activities for implementing in a school; thus they focused on 
thought knowledge. Participants reflected on the fact that they saw the cultural di- 
versities in students from different cultural backgrounds. These variations contrib- 
uted to the differences in students' responses in situations of school learning. The 
knowledge participants of Math Camp generated was "local knowledge of teach- 
ing" that could not be generated outside the practice (see e.g., Krainer, 2003, p. 
98). The prospective teachers indicated from these learning communities that they 



acquired knowledge that was not included in the elementary mathematics teacher 
education programme. 

The third issue concerns how learning happens in the group. This crucial issue 
is somehow related to the fourth issue that concerns the roles and relationships of 
the participants. The roles and relationships of participants in a community might 
change simply by making adjustments in their operation. Such changes are neces- 
sary in order to meet the members' interests and learning needs. Let us regard, for 
example, the Du Shu Hui for Advanced Mathematics and the Du Shu Hui for En- 
hancing Mathematics Undergraduate Understanding: Each of these two learning 
communities had one member that had a greater mathematical competence than the 
others. Both began with equal competence among the participants such that they 
criticized each other and helped each other solve problems, but evolved to a learn- 
ing activity which was dominated by a student superior in mathematical compe- 
tence taking over as the leader. This is most evident in the Du Shu Hui for Enhanc- 
ing Mathematics Undergraduate Understanding. Among the members, the junior 
student played the role as an instructor. What he learned during the process of in- 
struction is that he became aware of what he did not understand, which he then 
discussed with other members. This is how learning happened in a community with 
a definite leadership. 

In the case of Portugal, the learning communities were study groups of school 
courses and a research project. The prospective teachers in these communities were 
learning through an open and flexible discourse among them. For example, the 
study groups in the field-based course had no fixed curriculum, enabling a flexible 
planning of work with the contribution of all participants. And as the prospective 
teacher who participated in a research project described there was plenty of oppor- 
tunity to talk to others, know what they were doing, learn about their questions, and 
thus cultivate an inquiry stance. 

Learning may happen through reflecting on critical friends' comments in the 
interacting phase of a community's meetings. In the Du Shu Hui for examinations, 
prospective teachers started to learn together collaboratively. During the inter- 
views, some prospective teachers expressed that the closer to the days of examina- 
tion for teacher selection, the participants became aware that they were indeed 
competitors for the sparse number of teaching positions. However, they further 
expressed that this competitive relationship among participants did not change the 
operation within the learning community, particularly, the role of being a critical 
friend to one another on teaching demonstrations and in the analysis of instruction 
units. This regulation of operation showed that the relationship of critical friends 
between participants remained in the Du Shu Hui despite the competition for lim- 
ited teaching positions. 


The first author thanks Yuh-Chyn Leu, Yuan-Shun Lee and Wen-Xiu Xu for con- 
ducting interviews and Yu-Ping Chang and Jia-Rou Hsieh for preparing the manu- 
scripts of cases from Taiwan. 




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Fou-Lai Lin 

Department of Mathematics 

National Taiwan Normal University 


Joao Pedro da Ponte 

Departamento de Educacao da Faculdade de Ciencias 

University of Lisbon 









Studies on Their Professional Growth 

In this chapter we examine the research on and by mathematics teachers working 
together on their practice. We examine both groups of teachers within a school, 
that we are calling face-to-face communities, as well as wider networks across 
schools, in regions or even national initiatives. We give examples of research and 
we raise a range of issues that call for consideration when embarking on the use of 
such networks and communities. Of particular concern is the sustainability of such 
work beyond the engagement of teachers in research or professional development 
activities. By "sustainability" we mean the lasting continuation of achieved 
benefits and effects of a project or initiative even after its termination (DEZA, 
2005). We draw on the literature and our own experiences to suggest critical 
factors that may lead to sustainability. It may be the case, we suggest, that teachers 
need to engage in some type of research mode with issues that face them on a 
regular basis. The difficulties of sustainability might suggest more systematic 
support at state level for learning communities engaging in such activity in every 
school and across school networks. 


Researching practice in teacher groups within a school, when this takes place, may 
take a number of forms and serve different purposes and goals. It is perhaps not 
unusual to find mathematics departments in high schools wanting to analyse their 
student achievement data to identify areas for development. There are likely to be 
experiments with new resources that may be examined systematically. In general, 
though, we must say that teachers in most countries at all levels are under 
increasing pressure of expectations and demands, with performance management 
criteria, targets for students' achievements and other constraints. Time for 
systematic research by teachers is difficult to find, even when the will is there. It is 
therefore particularly important to identify what has and is being done in this area 
in order to inform others of the feasibility of such work and to provide some 
evidence of teachers' experience in researching their practice as members of a 

In this chapter, we will examine the literature on communities of practising 
mathematics teachers to identify the kinds of organisational structures that have 
been established, the issues that have been identified for development, the aims of 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 133-153. 
© 2008 Sense Publishers. All rights reserved. 


the initiatives, methodologies, and outcomes. Sustainability is a key issue in such 
initiatives, particularly where the initial motivation is for higher study or driven by 
a research project rather than the more intrinsic goal of improvement of teaching 
and learning per se. We will also discuss the sustainability of professional 
development projects and identify communities and networks as key factors 
fostering the long-term impact of such initiatives. 


Perhaps the most extensive framework for the study of teaching and learning is the 
Lesson Study (see also Yoshida, volume 1 ; Nickerson, this volume) which has been 
implemented in Japan for many years as a method for teachers' professional 
development and taken up more recently in the US (Stigler & Hiebert, 1999; Lewis 
& Tsuschida, 1998; Fernandez & Yoshida, 2004; Puchner & Taylor, 2006). There 
are also cases in which Japanese teachers provided support to US teachers to 
engage in Lesson Study (e.g., Fernandez, Cannon, & Chokshi, 2003). Briefly, 
lesson study involves a group of teachers in a school analysing in great depth how 
to improve specific lessons within particular topics, such that over a period of time 
the way concepts are taught changes. Initial development of a lesson will typically 
be followed by one teacher teaching the revised lesson, observed by the other 
teachers. They meet subsequently to review what happened, make further changes, 
and possibly teach the re-revised lesson again. 

Other studies in the US (see also e.g., Peterson, 2005) have investigated the 
impact of Lesson Study on prospective teachers and indicated that this can support 
growth in understanding of what to teach which can, in turn, lead to growth in 
understanding of how to teach (Cavey & Berenson, 2005). Puchner and Taylor 
(2006) found that the Lesson Study work can have an impact on teacher efficacy 
since teachers have the potential to discover through their lesson planning that their 
work does have an impact on their students' engagement and learning activities in 
class. In a major review of the potential of lesson study to contribute to 
instructional achievement Lewis, Perry, and Murata (2006, p. 3) suggest that we 
have few examples of full Japanese lesson study cycles on which to base research 
and development: 

Yoshida's (1999) dissertation case of mathematics lesson study in a Japanese 
elementary school (which formed the basis for Stigler's and Hiebert's chapter 
on lesson study and is now available in Fernandez & Yoshida, 2004); and a 
case of science lesson study in a Japanese elementary school [...]. 

Lewis, Perry, and Murata (2006) call for more detailed exemplars, which can 
help us try to understand the mechanisms and cycles of design-based research. The 
bibliography of review indicates, however, that whilst these deep studies are not 
yet available, a great deal of work is taking place using Lesson Study, in 
mathematics and other curriculum subjects. 

There is a long tradition of action research in education amongst practising 
teachers (see Krainer, 2006a; Benke, HoSpesova, & Ticha, this volume) with the 



aim of improving one's own teaching, drawing inspiration from Lewin (1948) and 
SchSn (1983). The goal of action research is to bring about change in one's own 
setting, be that classroom, department or indeed whole school (Stenhouse, 1975; 
Elliott, 1978). Generalisability is not a major concern, site-specificity is recognised 
as what matters. As such, the action research movement, which "can be traced back 
to Stenhouse's (1975, p. 142) reconceptualisation of curriculum development from 
an objectives to a process model" (Adler, 1997, p. 88) saw itself as, first, making 
educational research relevant to the classroom and, second, as giving teachers a 
voice in research, indeed the voice in research, as teachers were claiming power 
over what was researched and by whom, and the right to make the decisions over 
what the research was intended to change. It was, for some teachers, a response to 
the university researcher who came into their classroom, recorded some data, and 
then went away to publish in journals which no teacher read, in the furtherance of 
their career. There are strong links between action research and reflective practice 
(Leitch & Day, 2000), whereby in the act of teaching one reflects and acts, usually 
unconsciously, called reflection-in-action, and later perhaps one reflects on action 
(Sch5n, 1983), this latter leading to a recognition of the need to change something, 
coming therefore into line with action research. One might describe action research 
as reflection-o/j-reflection-0/7-action. It is intentionally set against an instrumental 
or technical view of action research. A search of the leading relevant journal 
Educational Action Research reveals very few studies in mathematics education 
although the approach is known within the community (e.g., Crawford & Adler, 
1996; Lerman, 1994). The two articles in the 2006 issue of the Journal of 
Mathematics Teacher Education by Goodell (2006) and Stephens (2006) draw on 
reflective practice and action research in their studies of prospective teachers' 
learning and their own practice, indicating that teacher educators can find action 
research a fruitful research orientation to support prospective teachers' learning. 
Goodell refers to the approach she took to her own practice as a teacher educator as 
self-study but this also has very similar features to action research, this latter term 
being more present in the mathematics education research literature (Schuck, 2002; 
Dinkelman, 2003). Stephens (2006) demonstrates the role of prospective teachers' 
reflections on their students' learning for developing awareness of their own 

To return to the way in which teachers' action research can be more than 
instrumental or technical, the revolutionary potential of action research was taken 
up by Carr and Kemmis (1986). They used Habermas's (1970) three-fold 
classification of social action: technical; practical; or emancipatory and they 
emphasised the need to take an emancipatory approach, which called for collective 
action to change education. However, action research is today often motivated by 
higher study goals or it may be instigated by engagement in research projects, 
usually in collaboration with university researchers. The sustainability of action 
research beyond the course of study or after the researchers leave, or if the funding 
runs out is clearly an issue, one we take up later in this chapter. 

We must note here also the external drivers for face-to-face researching 
communities. We have indicated that the regulatory systems of education in many 



countries have restricted teachers' freedom to engage in research in their 
departments on their pedagogy, curriculum, assessment or other aspects. 
Conversely, that same regulatory system in many countries (e.g., UK, the 
Netherlands) is requiring schools to produce development plans for school 
improvement and in most cases preparing those plans and acting on them devolves 
to departments. This can be an opportunity for subject-specific quality 
development and quality assurance that will have to be taken on board and acted 
upon by senior management in the schools. We report below on some research in 
this area. It may well be that a community may develop from the work called for 
by the regulatory system. 

In general, we might note that cooperation between mathematics teachers for the 
improvement of their pedagogy and their students' learning is possibly less 
common than in other school subjects because of the tendency to teacher autonomy 
(Lortie, 1975), particularly in mathematics. However, Visscher and Witziers (2004) 
examined the performance of school students in mathematics differentiated by how 
cooperatively mathematics departments in those schools operated, using a six-point 
scale for assessing the form and degree of cooperation. "A positive relationship 
was found between departmental policy, on the one hand, and student achievement 
on the other" (p. 796). 


To this point, we have set out some theoretical considerations in relation to 
teachers working together in face-to-face communities and networks to change 
mathematics teaching and learning. In this section we will present some examples 
of within-school research and across-school projects on mathematics teaching and 
learning. It is our intention to select examples that highlight key features of 
successful groups, "success" being defined locally, that is, by the participants 
themselves. It seems to us that it is in the spirit of the theme of this chapter that we 
do not judge particular initiatives by objectively developed measures, if such 
measures could be developed at all, but by the subjective evaluations of the 
participants, both university staff and teachers. We will "collect" together the key 
features from each of the examples and, following a review of the literature and of 
other initiatives, we will present them all, emphasising what, for us, are the most 
critical factors that enable the success of any programme to be sustained beyond 
the end of the specific project, namely the establishment of communities or 

In the first three examples, all of face-to-face communities, we have chosen 
ones that among them cover the main possibilities in terms of initiation of such 
networks. The first example was the initiative of a group of teachers in a school, 
the second is a mutual collaboration between university researchers and school 
teachers, and the third is the initiative of university researchers. Our fourth 
example sets the scene for the study and analysis of across-school networks which 



Example 1 

Arbaugh (2003) describes her work with a study group. A study group is defined as 
"a group of educators who come together on a regular basis to support each other 
as they work collaboratively to both develop professionally and to change their 
practice" (p. 141). A mathematics department chair in a US high school contacted 
the university researcher, Fran Arbaugh, for support in the department's 
development of a way of teaching geometry that was more student-centred and 
inquiry-based than their current practice. Over a period of six months the group 
met ten times, drawing on what is called the Mathematics Task Framework (Stein, 
Smith, Henningsen, & Silver, 2000) which "focuses on the levels of cognitive 
demand required by mathematical tasks and the various phases tasks pass through 
in their instructional use" (p. 142). The seven teachers who participated throughout 
reported that they valued the development of community and relationships that 
supported their learning. They particularly benefited from sharing experiences 
about aspects of their teaching, such as managing whole-class discussion, and 
appreciated the way the group worked, whereby no one person dominated, as is 
usually the case in other forms of education for practising teachers. They talked of 
the opportunity to think things through, to question and to experiment. They all 
spoke about how the study group enabled them to bring the theories, as used in the 
research community and practice together, facilitated by the research articles 
provided by Arbaugh. There is clear evidence of curriculum reform and change and 
a growing sense of their professionalism. One teacher commented (Arbaugh, 2003, 
p. 153): 

I'd not spent a lot of reflection time before. When I did, I mainly thought of 
how the kids could learn better. Now I look at things I could do to help them 
learn better. I look more at how I can create an opportunity for them to learn. 

Key features of the success of this activity, according to Arbaugh, were the 
financial support that enabled the teachers to be free from their teaching to hold the 
meetings and the fact that the teachers were all from the same school. Arbaugh 
highlighted the tension for herself between being the expert needed to introduce 
new content and the autonomy and empowerment that teacher groups needed to 
really benefit from working within a community. Arbaugh notes that the group 
continued alone for a further year to work on algebra. We find it interesting that 
there is no mention of action research in the article, even though the research seems 
to have followed that approach. Features of the activity not mentioned specifically 
by Arbaugh but important for this review are the focus on content-specific material 
and the opportunity for addressing pedagogical content knowledge. 

Example 2 

We have referred to Japanese Lesson Study above. A modified version, called 
Action Education, has been developed and used in China (Huang & Bao, 2006). 
These authors, in designing their approach, argued that the established and well- 



known benefits of Lesson Study can be, and need to be, enhanced by the 
participation of experts who can provide input on new content and on pedagogic 
issues, whilst the control of what is changed and improved in the mathematics 
lesson remains in the hands of the teachers. In their model, the process begins with 
a teacher teaching a lesson, with a focus on an issue that requires reflection and 
examination, in front of a group of other teachers and experts (Huang & Bao, 2006, 
p. 280): 

A fundamental feature of "Action Education" is that the unfolding of the 
program is mediated within the community by the whole process of 
developing an exemplary lesson (Keli), including the lesson planning, lesson 
delivery and post-lesson reflection, and lesson- re-deli very [...] a 
collaborative group (the Keli group) that consists of teachers and researchers 
is established through discussion between researchers and a group of 
interested teachers. 

Huang and Bao locate their approach within the literature on action research. It 
is a requirement of the whole learning process that the teacher group writes a 
narrative article that summarises their experiences, the changes, and their findings. 
The authors give an example of work on teaching Pythagoras' Theorem in which 
the group was formed from teachers in the school and a group of researchers from 
the local university. The teachers were aware that using an inductive approach to 
the proof tends to fail due to the problems of measurement, whilst expecting 
students to find a proof for themselves is unreasonable. They designed a series of 
lessons around the following diagram and the associated proof of Pythagoras' 

Figure I. Diagram for the proof of Pythagoras' Theorem. 

Using a series of worksheets building from numerical examples to the general, 
students' learning was scaffolded to a proof of the theorem. 

The findings of the study revealed a number of interesting and important 
changes. The first to be commented upon in the article is the change in what Huang 
and Bao (2006, p. 290) call the teaching paradigm: 

Through a comparison of the time distribution between the previous lesson 
and the revised lessons in terms of teacher talk, teacher-student interaction, 
student exploration and student practice, it was found that the time for teacher 
talk, and student practice went down from 51.2% to 26.7%, and from 28.2% 



to 3.2%, respectively, while the times for teacher-student interaction, and 
student exploration went up from 16.8% to 23.5% and from 3.8% to 46.6%, 

They argue also that students learnt not only the proof but also why it constitutes 
a proof. They go on to illustrate the teachers' reflections and hence their learning 
through Keli. In their conclusions, Huang and Bao (2006, p. 295) point to some of 
the problems faced by this approach to development for practising teachers: 

[...] where do the qualified experts come from? How to organize a practice 
community including teachers and researchers? How to re-schedule the 
teaching program for doing Keli without disturbing the normal teaching 
program? How to simplify the Keli model for use by teachers, so that the 
phases of the model can be easily understood and implemented effectively by 

They also emphasise the importance of the role of the expert: 

It also shows that the expert's roles and follow-up action are important for 
participating teachers' professional learning and changes in classroom 
practice (p. 295). 

Key features, beyond those mentioned by the authors, are the focus on both 
content and pedagogical knowledge and skills and an open, learner-centred 
implementation component. 

Example 3 

Drawing on a situated theory of learning, Kazemi and Franke (2004, p. 205) 
describe the work of a group of teachers across a year examining their students' 
work in an attempt to develop their understanding of their students' mathematical 
thinking. They work with a notion of transformation of participation in studying 
the change in the teachers over that time. 

The transformation of participation view takes neither the environment nor 
the individual as the unit of analysis. Instead, it holds activity as the primary 
unit of analysis and accounts for individual development by examining how 
individuals engage in interpersonal and cultural-historical activities. 

The intervention was the initiative of the university researchers and was based 
on pilot work using the Cognitively Guided Instruction (CGI) approach (Carpenter, 
Fennema, Franke, Levi, & Empson, 1999). The researcher, called facilitator in the 
paper, worked with the teachers to choose a common problem, modified by each 
teacher to suit her or his students. The group then met to discuss the strategies their 
students had used, and what those strategies revealed about their students' thinking. 
The researcher might add some findings from research, and assist the teachers in 
seeing commonalities and differences between their own findings and the research. 



Data were collected of the group meetings and in the classrooms. Kazemi and 
Franke (2004, p. 213) highlight: 

Two major shifts in teachers' workgroup participation emerged from our 
analyses. The first shift in teachers' participation centered around attending to 
the details of children's thinking. This shift was related to teachers' attempts 
to elicit their students' thinking and to their subsequent surprise and delight 
in noticing sophisticated reasoning in their students' work. The second shift 
in teachers' participation consisted of developing possible instructional 
trajectories in mathematics that emerged because of the group's attention to 
the details of student thinking. 

As Key Features for teachers' learning, Kazemi and Franke emphasise two 
factors in particular as mediators of teachers' learning: the role of the facilitator in 
guiding their reflections and appropriate use of the students' work as material upon 
which to reflect. Interestingly also, from our point of view, is the indication by the 
authors that the initiative was sustained beyond the year of interaction with the 
facilitator. From our point of view we would add: the focus on both content and 
pedagogical knowledge and skills; an open, learner-centred implementation 
component; and the prolonged duration of the activity. 

Example 4 

Background. The Austrian Ministry of Education put in place the project 1MST 
("Innovations in Mathematics, Science, and Technology Teaching"; see Krainer, 
2007; Pegg & Krainer, this volume). The project represents a nation-wide support 
system in the areas of mathematics, science and technology as well as related 
subjects. This project is working on the levels of professional, school, and system 
development, integrates systematically the principles of evaluation, gender 
sensitivity, and gender mainstreaming, and comprises several central programmes. 
One of these programmes is the setting up and supporting of Regional and 
Thematic Networks of teachers and schools. While the concept of Regional 
Networks is targeted at local and regional level, the Thematic Networks aim for 
cooperation between teachers across the whole nation. These Regional and 
Thematic Networks were successively established in 2004 and have gained 
importance in recent years, as they enable the economical use of available human 
and material resources. Beyond this, the establishment of such networks is 
expected to facilitate the setting of regional subject-related or cross-subject goals 
which direct the support in the teaching of mathematics, science and technology. 
Each Regional or Thematic Network is coordinated by a steering group, consisting 
of teachers, university staff, school authorities, and further relevant persons 
(Krainer, 2005). More than 8000 teachers have already participated in network 

The implementation of Regional and Thematic Networks aims at three goals: (a) 
enhancement of attractiveness and quality of teaching and school development; (b) 
professional development of teachers; (c) increased numbers of participating 



teachers and schools by enhancing regional and national communication structures. 
The design of the networks is based on several principles: (a) utilisation of existing 
human, institutional and material resources; (b) accountability of participating 
persons and organisations; (c) target-oriented thinking and acting together with 
systematic evaluation (balance of action and reflection); and (d) autonomous 
thinking and acting of persons or organisations in close interplay with the shared 
goals and principles (balance of autonomy and networking). In cooperation with 
school authorities and universities, teachers are implementing innovative projects 
aimed at these goals and principles (Rauch & Kreis, 2007). 

In the focus: The thematic geometry network Recent technological development 
and innovation processes show impact on the teaching and learning of mathematics 
as well as on teacher education (see e.g., Llinares & Oliveira, this volume; Borba 
& Gadanidis, this volume). For example, the increasing number of accessible 
internet resources and miscellaneous packages of CAD (computer aided design) 
software are changing the context of teaching and learning geometry. This change 
tends to result in several consequences. On the one hand, in Austrian secondary 
schools descriptive geometry is not part of mathematics and represents an 
independent subject. The development of new technologies leads to fundamental 
changes in the subject's curriculum, which in turn have to be implemented by the 
teachers. On the other hand, the implementation of these new contents and 
techniques requires highly skilled teachers. Even though many secondary 
mathematics teachers are highly engaged and innovative, they often lack 
knowledge and practice regarding these new issues, simply because in Austria 
hardly any descriptive geometry courses for teachers' professional development are 

To address these issues and to support the concerned teachers, the ADG (the 
professional association of geometry in Austria) decided to implement action by 
establishing the Thematic Geometry Network, a face-to- face network of practicing 
mathematics (in particular geometry) teachers and teacher educators interested in 
subject-didactics (Gems, 2007). This is seen as a step towards becoming an 
Austrian Educational Competence Centre (AECC). So far in Austria six such 
centres are established, among these is one for mathematics education. The 
geometry network was created in November 2005 and aims at improving 
communication between descriptive geometry teachers, quick and direct exchange 
of information concerning recent subject-related development, and the organisation 
of a database containing professional development programmes and expert pools. 
The spirit and guiding idea of the Thematic Geometry Network is characterised by 
shared intentions and goals, mutual trust and respect, voluntary participation, and a 
common exchange principle. 

On a concrete level, the network designs, initiates, implements, and evaluates 
various projects. All activities are coordinated by the network's steering group, 
consisting of teachers representing participating school types and levels. This 
group acts as both an inward central administrative network node (e.g., for 



planning meetings or distributing learning materials), and an outward contact point 
(e.g., providing information or handling public relations). The steering group 
together with different subgroups on national, regional, and local level frame the 
structure of the network (Mailer & Gems, 2006). The Thematic Geometry Network 
initiates and organises annual nation wide meetings of geometry teachers: Within 
the scope of these "subject didactics days" the teachers and working groups share 
information and ideas concerning geometry teaching and learning, curriculum, 
prospective teacher training, and professional development. 

Various working groups within the network deal with issues concerning 
geometry. One particular group prepares the content, curriculum, and 
implementation of a modified initial teacher training model. Another working 
group develops and provides support and assistance regarding 3D-CAD-software 
for students and teachers by designing and offering online tutorials and examples. 
A third group coordinates CAD modelling contests for students of secondary 
schools on regional and national levels. Yet another working group provides and 
maintains a touring geometry exhibition that contains objects, working and 
experimental stations, as well as concomitant information and material. In the 
future, the Thematic Geometry Network intends, on the one hand, to deepen and 
consolidate the actual projects and, on the other hand, to expand its activities by 
focussing on issues of competences, standards, assessment, and evaluation. In 
particular, by becoming a competence centre (AECC) the network could serve as a 
support structure for practicing teachers (Gems, 2007). 

There are two Key Features for the success of this network initiative. The first is 
the existence of teacher leaders that are willing to overcome the lack of an 
adequate support system for teachers' professional development. The second is the 
window of opportunity when IMST offered the establishment of Thematic 
Networks. In short: It is a combination of internal human resources and external 


Rationale for Examining Sustainability 

Most reform projects are initiated to enhance the quality of teaching and learning in 
the specific setting of schools and across regions and nations. Ingvarson, Meiers, 
and Beavis (2005, p. 2) state that 

Professional development for teachers is now recognised as a vital 
component of policies to enhance the quality of teaching and learning in our 
schools. Consequently, there is increased interest in research that identifies 
features of effective professional learning. 

In particular, the formation of face-to-face communities and networks, as 
described in the examples above, is seen as one of the most promising ways to 
reach the goal of enhanced teacher quality. In the next section, the following 



questions are addressed: 1) "What kinds of effects on teacher quality are possible?" 
and 2) "How can this impact be sustained?" 

The expected effects of such projects by both the facilitators and the participants 
are not only related to the professional development of individual teachers to 
improve teacher quality, but also to the enhancement of the quality of whole 
schools, regions and nations. Expected outcomes are not only focused on short- 
term effects that occur during or at the end of the project, but also on long-term 
effects that emerge (even some years) after the project's termination (Peter, 1996). 
The desideratum of all such projects and community building activities for 
providing teachers support and qualification is to enhance the learning of students. 
As Mundry (2005) states, "We recognize professional development as a tool 
focused on improving student outcomes" (p. 2); "Funders, providers, and 
practitioners tend to agree that the ultimate goal of professional development is 
improved outcomes for learners" (Kerka, 2003, p. 1). This strategy, to achieve 
change at the level of students (improved outcomes) by fostering change at the 
teachers' level (professional development and community building), is based on 
the assumption of a causal relationship between students' and teachers' classroom 
performance, "high quality professional development will produce superior 
teaching in classrooms, which will, in turn, translate into higher levels of student 
achievement" (Supovits, 2001, p. 81). Even though a variety of external factors 
influence students' outcomes (e.g., the socio-economic background of students' 
parents), the above mentioned hypothesis was verified in several studies such as 
Fennema and Loef (1992). Hattie (2003) states, "It is what teachers know, do, and 
care about which is very powerful in this learning equation" (p. 2). 

Most evaluations and impact analyses of professional development are 
formative or summative in nature; they are conducted during or at the end of a 
project and exclusively provide results regarding short-term effects. Apart from 
these findings which are highly relevant for critical reflection of the terminated 
project and necessary for the conception of similar projects in the future, an 
analysis of sustainable effects is crucial and the central goal of professional 
development; "too many resources are invested in professional development to 
ignore its impact over time" (Loucks-Horsley, Stiles, & Hewson, 1996, p. 5). But 
this kind of sustainability analysis is often missing because of a lack of material, 
financial and personal resources. "Reformers and reform advocates, policymakers 
and funders often pay little attention to the problem and requirements of sustaining 
a reform, when they move their attention to new implementation sites or end active 
involvement with the project" (McLaughlin & Mitra, 2001, p. 303). Despite its 
central importance, research on this issue is generally lacking (Rogers, 2003) and, 
"Few studies have actually examined the sustainability of reforms over long 
periods of time" (Datnow, 2006, p. 133). Hargreaves (2002, p. 120) summarises 
the situation as follows: 

As a result, many writers and reformers have begun to worry and write about 
not just how to effect snapshots of change at any particular point, but how to 
sustain them, keep them going, make them last. The sustainability of 



educational change has, in this sense, become one of the key priorities in the 

Levels of Impact 

When analysing possible effects of professional development, the question of 
possible levels of impact arises. Which levels of impact are possible and/or most 
important? How can impact be classified? Ball (1995) points out that teacher 
educators and facilitators should take a "stance of inquiry and experimentation" (p. 
29) themselves regarding these questions of impact. Recent literature provides 
some information as to the answers to these two questions; the following levels of 
impact are identified (Lipowsky, 2004): 

• Teachers' knowledge: This level includes different taxonomies of 
teachers' knowledge (e.g., subject knowledge and general and subject- 
specific pedagogical knowledge; e.g., Shulman, 1987), or attention-based 
knowledge (Ainley & Luntley, 2005), including knowledge about learning 
and teaching processes, assessment and evaluation methods and classroom 
management (Ingvarson et al., 2005). 

• Teachers' beliefs: This level includes a variety of different aspects of 
beliefs about mathematics, and its teaching and learning (Leder, 
Pehkonen, & Torner, 2002), as well as the perceived professional growth, 
the satisfaction of the participating teachers (Lipowsky, 2004), perceived 
teacher efficacy (Ingvarson et al., 2005) and the teachers' opinions and 
values (Bromme, 1997). Shifter and Simon (1992) highlight that change 
of teachers' beliefs is indeed common and desired, but is not necessarily 
an accomplished goal. 

• Teachers' practice: At this level, the focus is on classroom activities and 
structures, teaching and learning strategies, methods or contents 
(Ingvarson et al., 2005). 

• Students' outcomes: This level is related to professional development's 
central task, the improved learning and consequential results for students 
(Kerka, 2003; Mundry, 2005; Weiss & Klein, 2006). 

Classification and Analysis of Impact 

Classification and analysis of impact is based on two major types of effects: short- 
term effects that emerge during or at the end of a project; and, long-term effects 
that occur after the project's termination. Effects that are both short-term and long- 
term are considered by some to be sustainable. However, as Fullan (2006) points 
out, short-term effects are "necessary to build trust with the public or shareholders 
for longer-term investments" (p. 120). Although short-term effects are important 
and it may be that it is only possible to accomplish short-term impact, this does not 
provide for sustainable impact and the result would be to "win the battle, [but] lose 
the war" (p. 120), because sustainability, in this case, means the lasting 



continuation of achieved benefits and effects of a project or initiative beyond the 
termination of a professional development project or effort (DEZA, 2005). 
Hargreaves and Fink (2003) state, "Sustainable improvement requires investment 
in building long term capacity for improvement, such as the development of 
teachers' skills, which will stay with them forever, long after the project money has 
gone" (p. 3). Moreover, analysis of sustainable impact should not be limited to 
effects that were planned at the beginning of the project; it is important to examine 
the unintended effects and unanticipated consequences that were not known at the 
beginning of the project (Rogers, 2003; Stockmann, 1992). 

Factors Fostering Sustainability 

To give an overview and to summarise the literature concerning factors 
contributing to and fostering the sustainability of change, the following four 
elements of professional development projects are used to classify these factors: 
participating teachers, participating facilitators, the project or initiative itself, and 
the context that embedded the first three elements (Borko, 2004). Rice (1992) 
states that "these factors relate to the nature of teachers as people, schools as 
organisations and change processes themselves including many variables which 
facilitate or constrain change" (p. 470). Finally, we argue that the core factors 
fostering sustainability are community building and networking. 

Participating teachers. A professional development project should meet the 
teachers' needs and interests (Clarke, 1991; Peter, 1996) and it should be coherent 
with teachers' other learning activities (Garet, Porter, Desimone, Birman, & Yoon, 
2001), fit into the context in which they operate, and provide direct links to 
teachers' curriculum (Mundry, 2005). The teachers should be involved in the 
conception and implementation of the project; this allows teachers to develop an 
affective relationship towards the project by developing teachers' ownership in the 
proposed change (Clarke, 1991; Peter, 1996); it prepares and supports them to 
serve in leadership roles (Loucks-Horsley et al., 1996); and it focuses on the 
teachers' possibility to influence their own development process (empowerment) 
(Harvey & Green, 2000). These features act to facilitate attendance and trust, 
which in turn affects the teachers' future decisions concerning their development 
process. This ends up in a spiral process, which continually enhances the level of 
teachers' empowerment. 

An "inquiry stance" is another factor that fosters impact on sustainability 
(Farmer, Gerretson, & Lassak, 2003, p. 343; Jaworski, this volume). Teachers 
understand their role as learners in their own teaching process and try to 
understand, reflect, and improve their practice. This stance requires professional 
and personal maturity as well as the possibility to critically reflect one's own 
decisions and activities (Farmer et al., 2003). This notion was also used by 
Cochran-Smith and Lytle (1999, p. 289) when describing the attitude of teachers 
who participate in communities towards the relationship of theory and practice: 



"Teachers and student teachers who take an inquiry stance work within inquiry 
communities to generate local knowledge, envision and theorise their practice, and 
interpret and interrogate the theory and research of others". 

Participating facilitators. Similar to the teachers, are also the participating 
facilitators of the professional development programme. They also should take a 
"stance of inquiry" (Ball, 1995, p. 29) towards their activities, reflect on their 
practice and evaluate its impact (Farmer et a!., 2003). In addition, another 
important factor is the facilitators' knowledge, understanding, and whether they 
have a well-defined image of effective learning and teaching (Loucks-Horsley et 
al., 1996). 

Project or initiative. Each innovation, intervention, or project (e.g., the formation 
of communities and networks) is unique regarding its design. Therefore, impact 
analysis needs to include these differences in organisational and structural 
characteristics. Rogers (2003) studied the process of diffusion of innovations and 
pointed out that the impact of an innovation depends on several characteristics 
(Relative Advantage, Compatibility, Complexity, Trialability, and Observability). 
Fullan (2001) also described similar characteristics (Need, Clarity, Complexity, 
Quality and Practicality) that influence the acceptance and impact of innovative 

- Relative Advantage: This includes the individually perceived advantage of the 
innovation (which is not necessarily the same as the objective one). An 
innovation with greater relative advantage will be adopted more rapidly 
(Rogers, 2003). 

- Compatibility and Need denote the degree to which the innovation is perceived 
by the adopters as consistent with their needs, values and experiences (Fullan, 
2001; Rogers, 2003). 

- Complexity and Clarity indicate the adopter's perception of how difficult the 
innovation is to be understood or used (Rogers, 2003) which relates to 
concomitant difficulties and changes (Fullan, 2001). Thus, more complex 
innovations are adopted rather slowly, compared to less complicated ones. 
Clarity (Fullan, 2001) supports reducing complexity and perceiving advantages. 

- Trialability means the possibility of potential adopters to experiment and test the 
innovation on a limited basis. Divisible innovations that can be tested in small 
steps therefore, represent less uncertainty and will be adopted as a whole more 
rapidly (Rogers, 2003). 

- Quality and Practicality makes an impact on the change processes. High quality 
innovations that are easily applicable in practice are more rapidly accepted 
(Fullan, 2001). 

- Observability: The more the results of the innovation are visible to other persons 
(e.g., parents, principals) and organisations, the more likely the innovation is 
accepted and adopted (Rogers, 2003). 



For example, if face-to- face communities and networks are expected to make an 
impact, they should provide a high level of relative advantage, should meet the 
teachers' needs and their goals, and structures should be clearly and easily 
understandable, they should be practical and usable, of high quality and their 
results should be visible to others. 

Context. The particular importance of the school context that embeds participating 
teachers, participating facilitators, and the project or initiative itself, as a factor 
fostering the sustainability of innovations and change processes is well 
documented (e.g., Fullan, 1990; McNamara, Jaworski, Rowland, Hodgen, & 
Prestage, 2002; Noddings, 1992; Owston, 2007). Teachers promoting change (e.g., 
by participating in face-to-face communities and networks) need administrative 
support and resources (McLaughlin & Mitra, 2001). Support from outside the 
school by parents or the district policy is also an important factor (McLaughlin & 
Mitra, 2001; Owston, 2007). Moreover, school-based support can be provided by 
students and colleagues (Ingvarson et al., 2005; Owston, 2007), and in particular 
by the principal (Clarke, 1991; Fullan, 2006; Krainer, 2006b). As Owston (2007) 
states, "Support from the school principal is another essential factor that 
contributes to sustainability" (p. 70). Moreover, to foster sustainability not only at 
the individual (teacher's) level but also at the organisational (school's) level, 
Fullan (2006) proposes a new type of leadership that "needs to go beyond the 
successes of increasing student achievement and move toward leading 
organizations to sustainability" (p. 1 13), and calls these leaders "system thinkers in 
action". In particular, these school leaders should "widen their sphere of 
engagement by interacting with other schools" (p. 113) and should engage in 
"capacity- building through networks" (p. 115). 

Community building and networking as key factors fostering sustainability. The 
review of the literature on sustainability of structures for developing teaching and 
learning indicates that there is a great deal of research on factors that can lead to 
such initiatives continuing beyond the initiation stage. There is plenty of evidence 
to demonstrate that the community as a whole and the participants in particular 
benefit and learn from any initiatives to develop the teaching and learning of 
mathematics. That such projects may be initiated by higher level study, the 
research interests of university faculty or the immediate needs of teachers in a 
school or network of schools is the nature of the professional educational 
environment in most countries. 

Research findings, illustrated in large part by the examples given above, indicate 
that there are several characteristics of successful initiatives; we are particularly 
interested here in features that might foster the sustainability of a professional 
development programme or project: it should focus on content knowledge (Garet et 
al., 2001; Ingvarson et al., 2005), use content-specific material (Maldonado, 2002), 
and provide teachers with opportunities to develop both content and pedagogical 
content knowledge and skills (Loucks-Horsley et al., 1996; Mundry, 2005). 



Moreover, an effective professional development initiative includes opportunities 
for active and inquiry-based learning (Garet et al., 2001; Ingvarson et al., 200S; 
Maldonado, 2002), authentic and readily adaptable student-centered mathematics 
learning activities, and an open, learner-centered implementation component 
(Farmer et al., 2003). Further factors fostering the effectiveness and sustainability 
of the programme are: prolonged duration of the activity (Garet et al., 2001; 
Maldonado, 2002), ongoing and follow-up support opportunities (Ingvarson et al., 
2005; Maldonado, 2002; Mundry, 2005), and continuous evaluation, assessment, 
and feedback (Ingvarson et al., 2005; Loucks-Horsley et al., 1996; Maldonado, 

In particular, research indicates that communicative and cooperative activities 
represent key factors fostering sustainable impact of professional development 
programmes. This general result is supported by several authors and studies, even 
if the categories used to describe these activities are different: Clarke (1991), Peter 
( 1 996), and Mundry (2005) point to cooperation and joint practice of teachers, 
Loucks-Horsley et al. (1996) and Maldonado (2002) highlight the importance of 
learning communities, McLaughlin and Mitra (2001) identify supportive 
communites of practice, Arbaugh (2003) refers to study groups, and Ingvarson et 
al. (2005) stress professional communities as factors contributing to the 
sustainability of effects. In particular, providing rich opportunities for collaborative 
reflection and discussion (e.g., of teachers' practice, students' work, or other 
artifacts) presents a core feature of effective change processes (Clarke, 1991; 
Farmer et al., 2003; HoSpesova & Ticha, 2006; Ingvarson et al., 2005; Park-Rogers 
et al., 2007). In this regard, research findings, exemplified by some of the authors 
in our examples, point to the issue of power relations between the external 
facilitator(s) and the teachers in the school. There is no one prescription for the best 
form of such relations, but it is clear that teachers must feel themselves empowered 
and autonomous and that the expert(s) play a role that supports their work rather 
than dominates it. Once again, this issue recalls the motivation of the action 
research movement but with the emphasis on collaboration and cooperation, not 
one of excluding the academic. Both, internals and externals need to be regarded as 

Reducing the long list of key features, on the basis of the examples we have 
given above as illustrative of the field, we suggest that there is evidence to show 
that large scale, centralised projects across schools, such as the Thematic Geometry 
Network, are sustainable. Within-school developments such as lesson study are 
also sustainable provided that the developments are supported by school resources 
(time being the most important) and that they are seen by the teachers as what they 
choose to do for the improvement of their mathematics teaching and their students' 
learning, because they see the benefits and the impact. 

We believe that the job of a teacher should incorporate a lifelong process of 
learning about and developing one's teaching and two examples of sustainable 
structures, lesson study and Thematic Geometry Network, are indications of what 
works. Not all centralised interventions are benign of course, but where the 



participants (teachers in the main but also students and school managers), see the 
benefits of on-going research and development, sustainability is possible. 


Reviewing the work we have described and analysed in this chapter, we end here 
with some final, concluding observations: 

- There are few examples of either face-to-face or cross-school networks research 
when compared to the rest of the body of research on mathematics teacher 

- Where there are examples, the effects seem to be extremely positive on 
teachers' perspectives and understanding and, where there is evidence, on 
students' achievements. 

- Providing the opportunity for time out of the classroom for meetings may be a 
key to success but is costly. 

- Engagement of academics as experts appears to be crucial, at least because 
when teachers do research as a group without an academic it may not be written 
up and disseminated. In addition, teachers' groups often call for support in 
aspects of research activity, such as access to literature, research design and data 

- Community building and networking represent the core factors fostering 
sustainable impact of professional development programmes. 

- In order to go deeper with these observations, we unquestionably need more 
research on how fruitful initiatives can be sustained. 

- Whether dissemination of the outcomes and experiences of one network can or 
will be taken up by another network or by individual teachers remains an 
unresolved question. If it is the case that teachers' change and growth, perhaps 
what we should call teacher learning, only takes place when teachers engage in 
these activities themselves and not through reading the work of others, we may 
need to radically re-think and advocate that systematic national initiatives foster 
learning communities in every school as part of its work. 


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Stephen Lerman 
Department of Education 
London South Bank University 
United Kingdom 

Stefan Zehetmeier 

Institutfur Unterrichts- und Schulentwicklung 

University ofKlagenfurt 





Technologies, Interactions and New Forms of Discourse 

The chapter discusses the use of communication technologies and the development 
of new forms of discourse in the context of prospective mathematics teachers' 
learning. Two key ideas are identified from current studies and used to frame the 
processes that are generated when prospective teachers use new communication 
tools: the notion of community and the features of knowledge building processes 
and discourses, within a sociocultural framework. Three examples are discussed in 
details: creating and sustaining virtual communities and networks, constructing 
meaning through online interactions, writing and reading blogs and videopapers. 
Finally, the chapter draws together key factors that should be considered when 
computer-supported communication tools are introduced in mathematics teacher 
education and that seem to shape the characteristics of online interactions, 
construction of knowledge and creation and support of communities of practice. 


The use of information and communication technologies in Higher Education and 
in initial teacher education programmes has increased over the last few years. A 
range of new computer-based communication tools are now available for teacher 
educators to adapt and transform into pedagogical tools aimed at developing new 
approaches to teacher education (Blanton, Moorman, & Trathen, 1998; Mousley, 
Lambdin, & Koc, 2003). Communication tools are tools that typically handle the 
capturing, storing, and presentation of communication, usually written but 
increasingly including also audio and video. They can also handle mediated 
interactions between a pair or group of users. 1 Communication tools can be either 
synchronous or asynchronous, and include bulletin boards, e-mail, chats, virtual 
video-based cases, computer-mediated conferences and forums. These tools can be 
embedded in interactive learning environments that support the creation of virtual 
communities and networks. These new communication tools not only facilitate 
access to information, but also have the potential to change the personal and social 
relations amongst individuals and the way we understand the process of becoming 
a mathematics teacher (e.g., knowledge and identity, Borba & Villareal, 2005). 


K. Krainerand T. Wood (eds.), Participants in Mathematics Teacher Education, 155-179. 
© 2008 Sense Publishers. All rights reserved. 


Current studies explore a number of ways in which communication tools can be 
used in the context of teacher education. Nowadays, new technologies can be used 
to support interaction among prospective teachers. For example, computer- 
mediated communication may provide support for online discussions (Byman, 
Jarvela, & Hakkinen, 2005) during problem solving activities or for the creation of 
virtual communities. New communication tools such as forum and bulletin boards 
allow extension of classroom boundaries and provide opportunities to develop 
skills that might enable prospective teachers to learn from practice and develop 
knowledge-building practices (Derry, Gance, Gance, & Schaleger, 2000). Bulletin 
boards and online discussion also enable content-related communication including 
course materials, resources and activities and are used by prospective teachers as 
source of support for their development. Questions are generated about the social 
and cognitive effects of interactions in this online social space. In particular, it is 
interesting to look at how different forms of participation may operate towards 
mediating meanings in conversations within prospective teachers' learning. In 
these social interaction spaces, reciprocal understanding and the process of 
becoming a member of a community support the possibilities of taking and sharing 
different perspectives that require an understanding of others' points of view when 
they join in the same activity. By using new communication tools, new forms of 
discourse are emerging, as for example argumentative discourse, videopapers and 
blogs, which are used both as tools to represent and communicate knowledge, 
practice and research in a new form but also as tools for prospective teachers' 
reflection, self-reflection and assessment and for sharing good practice. 

The question of how these new forms of discourse and new forms of 
participation operate to mediate meaning construction in conversations and to 
create and sustain virtual communities is central to our understanding of the 
contribution of interactive learning environments and new communication tools in 
teacher education. In this sense, learning in collaborative settings is based on the 
assumption that learners engage in specific discourse activities and that learning 
stems from the relationship between the nature of the participation and the content 
of these discourses (learning is seen as becoming a member of a community that 
shares knowledge, values and skills). This situation assumes that computer- 
mediated communication is a tool used to mediate prospective teacher learning and 
reflective thinking and that it can mediate and transform teachers' experiences 
(Blanton et al., 1998). 

The introduction of new communication tools in mathematics teacher education 
is generating new research questions and is calling for new analytical procedures. 
What emerges from the literature is on the one hand the scarcity of research in this 
area, and, on the other hand, the attempts to use theoretical constructs from 
sociocultural perspectives of learning to explain the processes taking place when 
these tools are implemented. Common features to current studies are the 
description of the activities and assignments tackled by prospective teachers, the 
enumeration of the messages exchanged and the subsequent development of 
analytic categories drawing on discourse analysis (Schrire, 2006; Strijbos, Martens, 
Prins, & Jochems, 2006). Research also suggests that prospective teachers should 



learn how to contextualise these tools as learning means rather than using them in 
isolated initiatives. When these communication tools are used as mediators in 
prospective teachers' learning then some issues are generated about the nature of 
this kind of learning. Mathematics educators are attempting to provide claims 
about both the individual and collective construction of meaning in communities 
and on the relationship between discourse and knowing. Another important point 
for discussion is what role these tools play in relation to the specific nature of 
mathematical knowledge and knowledge of mathematics education. 

The chapter is structured taking into account these general features emerging 
from current studies on the use of tools and the creation of virtual communities and 
networks in the context of teacher education. In the second subchapter, we describe 
how the sociocultural perspective on mathematics teachers' learning is being used 
in current research studies. We identify two key ideas that can help us understand 
the generated processes: the notion of community and the features of knowledge 
building processes and discourses. These ideas underline different aspects of 
learning that are considered in those studies: learning as identity (becoming), 
learning as practice (doing) and learning as meaning (experience). Three examples 
of experiences with three different tools are introduced and discussed separately. 
Although we use a common theoretical framework to frame the three examples, 
each subchapter will highlight the specificities that each tool brings to the 
framework, as emerging from the literature. The third subchapter focuses on how 
virtual communities are created and sustained (learning as identity, becoming). The 
next subchapter deals with the question of how online interactions support the 
construction of meanings when prospective mathematics teachers are involved in 
solving specific learning tasks (learning as meaning, doing); and the fifth 
subchapter focuses on how the use of new forms of communication and discourse 
(blogs and videopapers) supports prospective mathematics teacher learning 
(learning as meaning, how experience becomes knowledge through discourse). 
Finally, in the last subchapter we discuss some emerging issues and suggest ideas 
for further research in mathematics teacher education. 


Sociocultural theories of learning and development offer useful conceptual tools 
for studying prospective teachers' learning when new communication technologies 
are introduced in mathematics teacher education. This perspective views learning 
both as a process of meaning construction and as a process of participation in 
mathematics teaching practices (Greeno, 1998; Lerman, 2001; Llinares & Krainer, 
2006). Sociocultural theories underline the social processes underpinning learning 
and consider that learning is mediated by participation in social processes of 
knowledge construction scaffolded by social artefacts or tools. These tools can be 
both technical tools and conceptual tools and are considered mediators of learning 
interactions in educational settings. Two notions appear to be essential when 
analysing the learning and development of prospective mathematics teachers: the 



notion of "communities" and the notion of "knowledge building" practices in 

Communities of Learning 

Communities of learning are formed by people who engage in a process of 
collective learning in a shared domain of human endeavour. The emergence and 
sustainability of some kind of community among prospective teachers seems to be 
an important mechanism in the process of becoming a mathematics teacher and in 
the transition from a university context to a professional context. The idea of 
community of practice introduced by Wenger (1998) in relation to learning in 
apprenticeship situations might be a useful analytical tool, but its translation to a 
context in which teaching is a deliberate process, as is the case in teacher 
education, is not an easy task (Graven & Lerman, 2003) and has fostered the 
necessary differentiation between communities, teams and networks (Krainer, 

One of the main features of the notion of communities of practice is that they 
are groups of people who share a concern for something they do and leam how to 
do it better as they interact on a regular basis. One relevant aspect in this 
characterisation is the notion of sharing a goal, as for example acquiring 
knowledge, skills and dispositions that are necessary to teach mathematics. An 
institutional context assumes the intentionality of learning and the existence of an 
expert or facilitator, but from a general perspective the definition of a community 
of practice according to Wenger allows for, but does not assume, intentionality or 
the existence of a facilitator. Although this constitutes a theoretical difference 
between how mathematics educators may use this notion and how this notion is 
used in other contexts, the construct of "community" provides new avenues that 
may help understand better prospective teacher learning and offers suggestions for 
teacher educators about how to design opportunities for learning. In the context of 
teacher education, learning is the reason why the prospective teachers come 
together, therefore teacher education programmes should define a shared domain 
of interest, as for example learning to analyse mathematics teaching in terms of 
student learning. In addition, this goal can also be considered as a framework for 
teacher preparation programmes that aim at helping prospective teachers learn how 
to teach from studying teaching (Hiebert, Morris, Berk, & Jansen, 2007), how to 
interpret classroom practices (Morris, 2006; Sherin, 2001), or how to conceptualise 
a contemporary view of mathematics teaching (Lin, 2005). So, at the very least, the 
intentionality should be explicit when mathematics teacher educators use this 
approach to think about the process of becoming a mathematics teacher and to 
design opportunities for learning. 

Three characteristics need to be fulfilled so that communities of practice emerge 
and are sustained and collaborative learning processes are knowledge productive 
(Wenger, McDermott, & Snyder, 2002): (i) a focus on shared interests and domain, 
(ii) the involvement in joint activities, discussions and sharing of information, (iii) 



the development of a shared repertoire of resources (experiences, stories, tools, 
ways to address recurrent problems). 

The first characteristic of a community of practice is the existence of a shared 
domain of interest that generates the idea of membership as a commitment to the 
domain. In the case of prospective mathematics teachers, we should see these 
conditions as part of the process of becoming a mathematics teacher, and so, of the 
process of generating ways of seeing the activity of mathematics teaching with a 
teacher's eye. Sometimes, this process is supported by reflections on the actions 
and experience of others through video-cases that encourage prospective teachers 
to participate in and reflect on discourse centred on mathematical ideas (Lin, 200S; 
Seago, this volume). In this process of learning to analyse teaching in terms of 
student learning, prospective teachers can generate a shared competence that 
characterises them as teachers. Hiebert et al. (2007, p. 47) conjectured the features 
of this domain in terms of four skills and knowledge "rooted in the daily activity of 
teaching, that when deployed deliberatively and systematically, constitute a 
process of creating and testing hypotheses about cause-effect relationships between 
teaching and learning during classroom lessons". 

Recently, information and communication technologies have been used to 
develop this characteristic of a community of practice. In the process of induction 
of primary mathematics teachers in professional communities, the constitution of 
an online mathematics community may provide both opportunities for sharing and 
communicating and access to quality resources (Dalgarno & Colgan, 2007; Goos 
& Benninson, 2008). 

The second characteristic of a community of practice is the way in which the 
members pursuing their interest in their domain engage in joint activities and 
discussions as a way of sharing information and building relationships that enable 
them to learn from each other. Analysis of mutual engagement amongst 
prospective teachers when they are solving specific tasks has pointed out that what 
is really important is identifying how, through mutual engagement, the prospective 
teachers define tasks and develop meanings for the different elements of 
mathematics teaching. The key characteristic "interact and learn together" is being 
introduced in the design of web-based learning environments (Wade, 
Niederhauser, Cannon, & Long, 2001), since it is considered that interaction and 
cognitive engagement during online discussion are critical for constructing new 
knowledge (McGraw, Lynch, Koc, Budak, & Brown, 2007; Zhu, 2006). From this 
perspective, communication tools can begin to mediate prospective teachers' 
thoughts, actions and interactions. 

Finally, the third characteristic of a community of practice is the development of 
a shared repertoire of resources, experiences, representations, tools and ways of 
addressing professional problems linked to mathematics teaching. Developing 
different ways to analyse teaching and to notice and interpret classroom 
interactions, or to interpret students' mathematical thinking is a process in which 
prior experience and beliefs are entwined. However, by using an instructional 
scaffolding process, it is possible that prospective teachers develop new ways of 
conceptualizing mathematics teaching. For example, Lin (2005) argues that the 



prospective teachers in his research, when constructing pedagogical 
representations, were able to articulate students' difficulties with a specific topic 
from multiple perspectives. 

New Forms of Discourse and Knowledge Building 

According to the sociocultural perspective about teacher learning (Wells, 2002), 
"knowledge building" has to do with ways in which prospective teachers are 
engaged in meaning making with others in an attempt to extend and transform their 
collective understanding. In this sense, knowledge building involves constructing, 
using and progressively improving different representational artefacts with a 
concern for systematicity, coherence and consistency (Garcia, Sanchez, Escudero, 
& Llinares, 2006; Llinares, 2002; Sanchez, Garcia, & Escudero, 2006). From this 
perspective, knowing is the intentional activity of prospective teachers who make 
use of and produce representations in a collaborative attempt to understand and 
transform their world (Wells, 2002). According to Wells (2002), the experience 
needs to be extended and reinterpreted through collaborative knowing, using the 
informational resources and representational tools of the wider culture, in our case 
mathematics education. 

Knowledge construction in collaborative settings is based on the assumption 
that learners engage in specific discourse activities and that the nature of the 
participation and content of this discourse is related to the knowledge thereby 
constructed (Sfard, 2001; Wells, 2002). We use and adapt here the notion of 
Discourses as developed by Gee (1996, p. viii), according to whom Discourses are 
"ways of behaving, interacting, valuing, thinking, believing, speaking, and often 
reading and writing that are accepted as instantiations of particular roles (or types 
of people) by specific groups of people". 

Prospective teachers might create points of focus around which the negotiation 
of meaning and reciprocal understanding become organised by generating 
processes such as noticing, representing, naming, describing, interpreting, using 
and so on, what Wenger (1998) calls reification. The process of reification shapes 
the prospective teachers' experience of creating "objects" about mathematics 
education that they use to notice and interpret mathematics teaching and learning. 

Some research has shown that the construction of knowledge and the 
development of the skills needed in order to generate a more complex view of 
teaching is a process in which the interrelationship and integration of ideas about 
teaching and learning are progressively included in the analysis and reflection by 
prospective teachers (Garcia et al., 2006; Sanchez et al., 2006; van Es & Sherin, 
2002). These ideas are viewed as "tools to think about" and handle mathematics 
teaching and learning situations. The progressive use of theoretical ideas as 
conceptual tools in activities of analysing and interpreting teaching and learning 
situations, and the progressive modification in the type of participation in the 
spaces set up for social interaction are manifestations of the knowledge 
construction process (Derry et al., 2000). Here we are paying special attention to 
the activity of knowing through making and using representational artefacts (e.g., 



the theoretical information provided in online discussions) as a means of guiding 
joint action and of enhancing collective understanding (as can be seen in the 
creation of a videopaper). The new interaction and communication tools (such as 
virtual debates, bulletin board discussions, videopapers, blogs) contribute to 
generating an ongoing discourse amongst prospective teachers that enables 
viewing "the said" as a knowledge artefact that contributes to the collaborative 
knowledge building of the participants in the activity. These new forms of 
discourse generated by the currently available communication tools use "writing" 
as an instrument for collaborative reflection and as a tool for inquiry. The new 
"type of text" generated by these communication tools can be used as an 
"improvable object" that favours the generation of a progressive discourse that acts 
as the focus of collaborative knowledge building (Wells, 2002). 


The use of communication technologies in teacher education has seen an attempt 
by researchers to develop a view of teacher learning as a social and cultural 
phenomenon. Within this framework, some teacher educators are now studying the 
role played by virtual learning communities through the use of electronic 
discussion boards. In particular, recent studies look at how these new 
communication tools support professional reflection, how communities are 
established and supported through online and face-to-face interactions, and what 
type of support an online community formed by prospective teachers and 
practising teachers can provide to prospective teachers (Dalgarno & Colgan, 2007; 
Goos & Benninson, 2008; Schuck, 2003). 

Schuck (2003) argues the role played by computer-mediated conferencing (e.g., 
electronic conferencing boards 2 ) in challenging prospective mathematics teachers' 
beliefs and poses the question of how teacher educators should express an opinion 
or suggest a course of action. In Schuck's study, a group of prospective primary 
teachers were encouraged to post questions to a forum, either about the use of 
technology or about the content of the course they were studying, with the aim of 
developing their understanding of mathematics and of mathematics teaching. The 
analysis of how the prospective teachers used the forum and of the content of the 
messages posted in the discussion board showed that the use of the forum to 
achieve this objective was irregular and that the role played by the students varied. 
Although there were prospective teachers who did not participate in the forum, for 
those who used the forum the discussions were useful to encourage reflection, to 
share teaching experiences without having to be on campus, and they also 
encouraged the process of justifying and explaining points of view. Because of 
these reasons, Schuck argues that accessibility to a forum is an important factor in 
developing a community of learners. This author raises the issue of whether 
participation should be compulsory due to the benefits identified, but in the end 

Electronic conference boards are web-based conferencing tools. 



proposes that the reasons not to participate should also be respected. In this last 
case, the prospective teachers should value the participation in the forum as an 
alternative way of learning. Another issue to take into account is the level of 
structure imposed on the use of and participation in bulletin boards. Questions that 
emerge from the study are: what conditions may restrict the use of bulletin boards 
and the free exchange of ideas, what conditions may determine the emergence of a 
community and what should be the role of mathematics teacher educators in 
supporting or suggesting new avenues. 

Also working with prospective primary teachers, Dalgarno and Colgan (2007) 
examine what is the support provided by an online mathematics community to 
prospective teachers. In their study, a group of prospective primary mathematics 
teachers sought opportunities to continue their professional development through a 
forum that they could access once they were out in schools after graduation. The 
needs identified by the prospective teachers were: discussion with experts, access 
to suggestions and help about mathematics content and to mathematical resources 
selected by the experts for use in the classroom (repository of exemplars and 
resources), having a place where they could share lesson plans and activities that 
they themselves had created (repository of novice teachers' ideas about teaching 
and learning). An online community called Connect-ME was created and was 
constituted by prospective teachers and beginning teachers. Connect-ME offered a 
means to meet the expressed needs through mixed-method delivery mechanisms. 
According to Dalgarno and Colgan (2007, p. 12), the online community Connect- 
ME provided novice teachers "with a safe, communicative community for sharing 
resources and ideas and an environment where they can proactively seek the help 
they need". 

This also highlights the significance of emotional and personal connections. 
Dalgarno and Colgan suggest three essential elements that help sustain this type of 
connections: (i) the initial community members should have a personal link to, and 
a loyalty and respect for, the project facilitator; (ii) the facilitator should continue 
to communicate with all members of the online community even after they 
graduate; and (iii) the online forum should be created and developed at the 
"grassroots level", but its growth ought to be the results of previous personal 

Goos and Benninson (2004, 2008) also study the interface between secondary 
school mathematics prospective teachers and beginning teachers and how such 
type of community is established and maintained through online and face-to-face 
interaction. These researchers look at an online community of practice established 
via Yahoo! Groups 3 with the aim to: encourage sessions and professional 
discussion outside class times and during the practicum periods; provide 
continuing support, by remaining accessible to its members after graduation. One 
characteristic of this website was that the authors imposed minimal structure on 
communication. Goos and Benninson are in fact interested to know how and why 




prospective teachers and beginning teachers might choose to use this form of 

One participation structure linked to the bulletin board that Goos and Benninson 
considered interesting was the fact that prospective teachers used the bulletin board 
to organise and negotiate the agenda of a debriefing session to take place after they 
returned to university. Prospective teachers from different cohorts, beginning 
teachers and teacher educators attended this debriefing session, discussing 
pedagogical challenges, identifying sources of assistance, and comparing the 
effectiveness of different teaching approaches. Afterwards, the bulletin board was 
used to provide a summary of the debriefing session for those who had been unable 
to participate. This internship debriefing session was organised the following year 
too, which is what Goos and Benninson interpret as the beginning of a professional 
routine and a part of the shared history of the community. 

Goos and Benninson (2004, 2008) also suggest a possible factor that might have 
influenced the creation and sustainability of this community, for example, their 
own role in shaping the interactions between the participants by offering models of 
online professional exchanges through forwarding messages from other e-mail 
discussion lists used by mathematics teachers, encouraging prospective teachers to 
share teaching resources and their mathematics teaching experiences. 

The studies mentioned in this subchapter start shading light on the processes of 
creating and sustaining communities of practice through online environments. Four 
factors that emerge are: (i) the provision of accessible and flexible online forums, 
discussions and bulletin boards which can be appropriated and adapted to satisfy 
the teachers' needs; (ii) the participation of both prospective and practising 
teachers to the same community, together with "experts", which may enable the 
construction of professional knowledge and practices, together with the creation of 
a shared repertoire of mathematics resources; (iii) the co-existence of online and 
face-to-face interactions, which also enables the creation of emotional and 
personal connections that foster continuous participation in the exchanges and 
discussions and the development of a shared history; and (iv) the provision of 
models of professional exchanges and interactions to get the teachers started and 
provide an initial structure for the discussion. 


The question of how forms of participation operate to mediate meanings in 
conversations is central to our understanding about the role of interactive learning 
environments in teacher education. Research on knowledge construction in 
collaborative settings is based on the assumption that learners engage in specific 
discourse activities and that the nature of the participation and the content of these 
discourses are related to knowledge construction (Llinares, 2002; McGraw et al., 
2007; Santagata, Zannoni, & Stigler, 2007). From the point of view of 
sociocultural perspectives on learning, it is assumed that prospective teachers 
construct arguments in interaction with their partners in order to build knowledge 



about mathematics teaching, as well as to develop the skills needed to learn from 

Llinares and his colleagues designed several learning environments considering 
this theoretical perspective and using "design experiments" as a methodological 
approach (Callejo, Vails, & Llinares, 2007; Cobb, Confrey, DiSessa, Lehrer, & 
Schauble, 2003; Llinares, 2004). Figure 1 displays the web-structure of one of 
these learning environments integrating video-clips of mathematics teaching, 
asynchronous computer mediated discussion, theoretical information related to the 
given task and links to written essays about mathematics teaching (Vails, Llinares, 
& Callejo, 2006). This particular design gives prospective teachers the opportunity 
to engage in the process of meaning making with other colleagues, in an attempt to 
extend and transform their collective understanding in relation to some aspects of a 
jointly undertaken activity. Video-clips or teaching vignettes are used to situate the 
individual cases in the context of classroom practices. In addition, this web- 
learning environment provides prospective teachers with theoretical information 
and questions aimed to generate online discussions and to promote an inquiry 
orientation towards the observation of mathematics teaching (which is one of the 
objectives of teacher education). 



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Figure I. Structure of the web-based learning environment integrating video-clips, 

theoretical papers, social interaction spaces (online debate) and possibility of writing 

essays about mathematics teaching. 



The adopted approach creates a setting in which prospective teachers come 
together in a virtual collaborative action and interaction setting. The website allows 
them to watch videos and to download reports in text format (transcription of the 
lesson in the video-clip, the activities used by the teacher during the lesson, and 
documents with theoretical information about the characteristics of mathematics 
teaching), at any time and from anywhere by logging in the learning environment. 

In the study described in Llinares and Vails (2007), the prospective primary 
teachers had the chance to observe aspects of a mathematics lesson from different 
perspectives, starting from their own initial conceptions (based on their experience) 
and moving onto positions in which they use conceptual elements introduced in the 
training programme (theoretical information). The progressive use of the 
conceptual tools in the online discussions about the analysis and the interpretation 
of the teaching and learning situations and the progressive change in the way 
students participated in the spaces set up for social interaction, are manifestations 
of the development of the skills needed to learn from practice and of the 
knowledge construction process. What was emphasised in the design of the 
learning environments in Llinares and Vails' study was the activity of knowing 
through making and using representational artefacts (the provided theoretical 
information) as means of guiding prospective teachers' joint action and enhancing 
collective understanding through online discussions. The relationship between 
online interaction and the construction of meaning is rooted in the assumption that 
the semiotic process through which the ideas are formulated and communicated 
towards the achievement of a goal during an activity is part of the construction of 

This organisation of the learning environments attempts to reflect the 
progressive and evolutionary nature of the process of construction of the 
knowledge needed for teaching (Goffree & Oonk, 2001) and of the required skills 
to learn from practice, in which conceptual tools are progressively integrated in the 
activities of analysis and reflection (Garcia et al., 2006). In this web-learning 
environment, when prospective primary teachers contributed to the online 
discussion they had to interpret the preceding contributions and to formulate a new 
contribution extending, questioning or qualifying what had already been said. Such 
progressively generated discourse mediated knowing. For example, in the 
exchanges below the prospective primary teachers refined and amplified previous 
contributions about how the idea of equity should be understood in the classroom 
context, when they analysed a video-clip in which a primary teacher managed the 
interactions among primary students posing problems from a commercial brochure. 

I do not entirely aeree (LUISA - 12:05:00 09/01/2006) 

I partly agree with the comments made by my colleagues Estefania and 
Angela, but although all the students participate and interact, the teacher did 
not carry out her task properly. The role of the teacher in the classroom is to 
get everyone to share while working on the problem (which I think she does, 
like my colleagues). But the teacher, in my opinion, does not pay attention to 



what the children say, nor does she show any real interest in their 
suggestions. During the class, two other children offer possible data to help 
in solving the problem. One boy says that there are 14 chocolates in the box, 
and another says there are 50. The teacher ignores these two children's 
suggestions, and proceeds to solve the problem using only the supposed 
existence of 8 chocolates. In my opinion she should have paid more attention 
to these two children and should also have solved the problem using their 
data. Not doing the exercise in this manner means that she does not respect 
the principle of fairness, because she has not listened to the contributions 
made by two of the children. 

To LU1SA (ANGELA - 14:28:06 09/01/2006) 

In one way you are right. The teacher does not accept all the children's 
proposals for the problem, she only accepts that of the child who suggests 8 
chocolates. At one point in the video the teacher says "There could be 8 
chocolates, or there could be more [...]" and "8 chocolates is a number which 
might be the one, or it could be different [...]", so 1 think that by saying that, 
she is trying to show that she has paid attention to the other opinions, but as 
she can't put all of them on the blackboard she decided to concentrate on the 
number proposed by one of the children, simply in order to set up the 
problem and solve it. I think that if the video were to continue and the teacher 
were to approach the problem again she would pay attention to a different 
child, and that if she required more data, she would use those suggested by 
several different children. 

The interactions amongst the prospective primary teachers, motivated by other 
students' contributions, were increasingly focused - in this case on the notion of 
equity as a valued-added dimension in teaching that can promote understanding - 
indicating that the structure of this type of environments, including the contexts 
and the activities, seemed to encourage the prospective primary teachers to engage 
in meaning making with others in an attempt to extend and transform their 
collective understanding of mathematics teaching. 

The study confirms that interaction occurs among prospective primary teachers 
when it is generated by a given task, which in this case was the analysis of 
mathematics instruction and of its effects on children's mathematical competence. 
The tasks and online discussions were intentionally integrated into the course in 
order to lead to a high degree of interaction. Llinares and Vails (2007) argue that 
the focus on shared interest helped the prospective teachers to engage in the joint 
activities of identifying and analysing different aspects of mathematics teaching 
thus enabling them to construct a shared understanding of the situation. 

The activities that were designed within these online learning environments 
required the prospective primary teachers to identify key aspects in a mathematics 
lesson and interpret them, something that seemed to encourage interaction. Here 
the role of the theoretical information was to help prospective teachers to begin to 



"notice" (Pea, 2006). From Wells' (2002) theoretical perspective, the process of 
knowledge-building is based on the assumption that students are engaged in 
specific discourse activities related to knowledge acquisition through discussions 
in which they focus on issues directly related to their future teaching. The 
interactions in the online discussions showed how the prospective teachers' initial 
personal interpretations could progressively be modified to construct common 
knowledge, when they perceived the ideas about mathematics teaching as 
functional in relation to the task that they had to undertake. 

These findings suggest that the different types of task and conditions of online 
discussion in the learning environments in which prospective primary teachers 
participated (the discussion questions and the video-clips) seem to exert an 
influence on the nature of the interaction (Schrire, 2006). The results of Llinares 
and Vails' analysis imply that the degree of prospective teacher involvement in 
interactive processes is related to the type of task intended to justify their 
participation in online discussions. Different structural factors that seemed to 
contribute to the construction of meaning were clear established goals with 
thematic prompts, and time to write. 

The existence of clear goals with thematic prompts seems to support the 
hypothesis that prospective teachers should use online discussions as a tool in their 
learning environments. That is to say, prospective primary teachers should 
consider participation in online discussions to be useful and beneficial for carrying 
out assigned tasks. The findings suggest that prospective primary teachers could 
identify in the task a focus on a shared interest - the goal - that justified their 
engagement in joint activities, and considered the online discussions as social 
spaces in which it was possible to develop a shared repertoire of experiences, tools 
and ways of addressing the analysis of videotaped case studies (Wenger, 1998). In 
this context, the personal interpretations were questioned and clarified in the online 
discussion, and assumptions and inferences were challenged in an attempt to 
construct a communal answer supported by common knowledge. This process was 
given further importance by the fact that the progressive discourse was conducted 
in writing. 

The questions for discussion that were set in the learning environment led the 
prospective primary teachers to respond to each others' messages, agreeing or 
disagreeing about different points of view. This enhanced their ability to see things 
from another's viewpoint and they began to develop a more complex view of 
teaching, as could be inferred from messages that became more focused on the 
specific topics over time (Byman et al., 2005) showing that writing was used as a 
tool for collaborative reflection where the text of the messages acted as an 
"improvable object" to the focus of collaborative knowledge building (Wells, 

In another context, McGraw et al. (2007) uses discussion prompts to stimulate 
critical analysis in a multimedia case and facilitate online discussions. In the 
project described by McGraw at al., online forums were used to discuss a 
multimedia case amongst prospective mathematics teachers, practising 
mathematics teachers, mathematicians and mathematics teacher educators. The 



analysis of the messages posted in the online forums and of the transcripts of the 
face-to-face discussions identified episodes of dialogic interaction in which 
individuals explicitly responded to the ideas and opinions of the previous writers. 
McGraw and her colleagues suggest that integration in discussion groups of 
members with different perspectives and level of experience enabled the 
generation of multiple episodes of dialogic interaction in each discussion group. 
Moreover, the interplay between theoretical and practical knowledge as a 
manifestation of knowledge building was evidenced by movements in the 
discussions between case specific observations and more general observations or 
use of theoretical knowledge. In relation to this last aspect, the variations in level 
of noticing in the different members of the group - prospective teachers, practising 
teachers, mathematicians, teacher educators - seemed to influence the development 
of know I edge. 

Summing up, two characteristics emerge as relevant as concerns online 
interactions and knowledge building: (i) providing structured guidance through 
tasks and discussion questions with thematic prompts seems to enable the 
participants in online discussions to reflect on and integrate multiple aspects of 
teaching; (ii) prospective teachers might benefit from engaging in discussions with 
more knowledgeable persons, grounded in a case of classroom practice, that can be 
accessed through an online environment from anywhere at anytime. The 
interaction amongst the multiple perspectives that may emerge, and be made 
explicit in the written messages appearing in the online discussions, from people 
with different levels of knowledge while "seeing" the same multimedia case, is 
vital to meaning making and knowledge construction. 


The relationship between the forms of discourse and knowledge building needed to 
become a mathematics teacher and the emergence of communities when new tools 
such as blogs and videopapers are used in mathematics teacher education have 
recently become an object of research (Beardsiey, Cogan-Drew, & Olivero, 2007; 
Makri & Kynigos, 2007; Nemirovsky, DiMattia, Ribeiro, & Lara-Meloy, 2005; 
Olivero, John, & Sutherland, 2004; see also Borba & Gadanidis, this volume). 

Blogs are (personal or organisational) web pages organised by dated entries 
whose items are links, commentaries, papers, personal thoughts and ongoing 
discussions. Blogs are considered as learning spaces with digital, sharable, and 
reusable entities that can be used for learning and are available to prospective 
teachers anytime and anywhere (learning objects). Recently, research interest has 
emerged about the potential and possible roles of blogs in the professional 
development of mathematics teachers and about the necessary relation between the 
use of these new tools of communication and the intentional ity of their use in 
mathematics teacher education (Makri & Kynigos, 2007). Makri and Kynigos 
focus their research on the ways in which prospective teachers write about both 
their subject and its pedagogy and study the discourse that is developed in 



teachers' blogs. In Makri and Kynigos' course the prospective teachers were given 
writing tasks addressing their epistemological and pedagogical beliefs and their 
subject related knowledge (e.g., the pedagogical value of using software in 
mathematics). The prospective teachers were encouraged to publish their answers 
on a blog and comment on the work of their peers by sharing opinions and 
engaging in discussion. 

The prospective teachers in this study used the explanatory and expository 
genres and their writings showed a structured cognitive presence since they 
combined factual knowledge and conceptual and theoretical knowledge which 
emerged collaboratively. The researchers suggest that the use of the blog 
introduced changes in the social orchestration of the course at an affective, 
interactive and cohesive level. Although research of this type is still recent, Makri 
and Kynigos identify different profiles of prospective teachers in relation to the 
emerging forms of social interaction indicating the degree of appropriation of the 
blog by the prospective teachers. The three profiles identified, blog enthusiasts, 
blog frequent visitors and blog sceptics, indicate that it is necessary to study in 
depth the changing social practices and roles and the new role of the instructor, as 
it has also been pointed out by studies on creating and sustaining communities of 
practice (Schuck, 2003). The sociocultural perspectives of learning assume that the 
social context affects the nature of learning activities, so when the social context is 
modified by introducing new forms of social interaction on the web it is assumed 
that this will influence the capacity to engage prospective teachers in collaborative 
activity, reflection, knowledge sharing and debate. 

Besides tools that facilitate social interaction through writing, such as blogs, 
other multimodal tools that integrate different forms of discourse in the same 
environment, such as videopapers, are beginning to be investigated in the context 
of teacher education. Videopapers (for an example, see Figure 2) are multimedia 
documents that integrate and synchronise different forms of representation 
including text, video and images, in one single non-linear cohesive document 
(Nemirovsky et al., 2005; Olivero et al., 2004). 

Combining the video with the text in a videopaper creates a fluid document that 
is more explicit than the text or video alone, while remaining contained and 
controlled by the author. Since their initial development in 1998 as an alternative 
genre for the production, use, and dissemination of educational research, research 
has investigated their potential and use in a variety of contexts ranging from 
teacher education to professional development to research collaborative practices 
(e.g., Barnes & Sutherland, 2007; Beardsley et al., 2007; Galvis & Nemirovsky, 
2003; Nemirovsky, Lara-Meloy, Earnest, & Ribeiro, 2001; Smith & Krumsvik, 

4 Videopapers are created with the free software VideoPaper Builder 3 (Nemirovsky et al., 2005), 
downloadable from 



v- *? ^ •* 



id wirti m Ik f am *■ Im4 irwww* anv««4 Mrf •■> MMA MMT« Of «MWgM)Ort •■ 

mhwWi»&« i » h fi— ) 



Figure 2. Screen shot from a videopaper. 

Considering videopapers as "products", Olivero et al. (2004) discuss the role 
they might have in teacher education and how they might support the 
representation and development of communities of practice where new ideas can 
be expressed and experienced. Reading about new and innovative approaches to 
teaching and learning can influences prospective teachers' beliefs and principles 
and impact on the process of "becoming a teacher", but reading does not provide 
any re-assuring image where the action is modelled vicariously. Moreover, 
prospective teachers might also lack the confidence to experiment if they have few 
realistic models to work from. Videopapers can provide prototypical instances of 
(innovative) practices combined with a range of supporting warrants. The study 5 
reported in Olivero et al. (2004) uses videopapers as a way to represent and 
communicate innovative uses of ICT in mathematics teaching, based on research 
findings, to both prospective and practising mathematics teachers. After reading 
one of these videopapers, a teacher typically comments: 

What is going on in classrooms is being communicated and it does make the 
project look real, real pictures in real classrooms. Seeing it, helps you to 
make sense out of it and gives you real models that you know are more than 
just tips for teachers. 

Such videopapers were seen as "more than tips for teachers" as they afforded 
the opportunity to develop a different kind of knowledge for teaching - knowledge 
not of "what to do next", but rather "knowledge of how to interpret and reflect on 

' More information about the project can be found in Sutherland et al. (2004). 



classroom practices" (Sherin, 2004, p. 17), because of the interplay between 
theoretical and practical knowledge they incorporated. Similarly to Vails et al. 
(2006), the teachers appreciated the theoretical knowledge embedded in the 
videopaper as a tool that could help them "notice" relevant practices, meanings, 
knowledge in the video: 

I have shown this to a number of teachers in Bristol - maths teachers - 1 was 
working with, they wanted something stimulating on proofs. So I did it but 
they wanted the bit on the research reading and the thinking cut out because 
they thought teachers would find it irrelevant and too time consuming. That's 
a problem because the thinking and the discussing and reading were so 
important. They are missing out [...] well sort of missing out parts of the 

Another teacher continues explaining how he thinks this kind of videopapers 
can be useful to teachers, who, thanks to the videopaper, would benefit from the 
results of the research project, in which the teachers collaborated with the 
researchers, even without being part of it: 

So they [teachers] would benefit I would hope from coming into my 
classroom, well we can't afford that because we can't get supply cover so they 
can watch my classroom instead [through the video in the videopaper]. They 
would benefit from talking to Tim [researcher], well they can't do that but 
they can understand what he is trying to say [through the text in the 
videopaper]. And they'd benefit from having my lesson plan which we 
worked on but they can adapt it for themselves. So they have everything - 
even the research evidence backing up the principles and practice. And also 
they can see the outcome, which is very important. If you do this they would 
do this, and this is the reason why we're doing it, this is me doing it and this 
is what they produce at the end. 

This quote suggests how in this case the videopaper is a tool that may afford the 
creation of and provide a space for the representation of communities of practice 
that bring together researchers, practising teachers and prospective teachers. 

Another project (Armstrong et al., 2005; Barnes & Sutherland, 2007) looks at 
the creation of videopapers to represent a collaborative research process in which 
both researchers and teachers interpreted classroom episodes within a series of 
mathematics and science lessons. Similarly to what McGraw et al. (2007) found in 
relation to the discussion of multimedia case studies, having people with different 
knowledge and expertise looking at the same data provides the possibility for 
different understandings of a lesson to be put forward and productively interact to 
create knowledge. The subsequently created videopaper embodies this process, 
enabling the multiple perspectives to coexist and to be grounded in the reality of 
the classroom (Galvis & Nemirovsky, 2003). While these projects see the process 
of creating a videopaper as an individual process, Zahn et al. (2006, p. 738) 



describe a pilot project on collaborative learning through advanced video 
technologies and analyse the use of the DIVER system 6 by prospective teachers: 

DIVER is based on the notion of a user diving into videos, for example, 
creating new points of view onto a source video and commenting on these by 
writing short text passages. Diving on video performs an important action for 
establishing common ground that is characterised as "guided noticing". 

The "diving" process can be shared and collaboratively developed. 

Although videopapers are mainly seen as objects or "products", research has 
shown that what is also important in terms of the development of prospective 
teachers' knowledge are the phases of actively creating a videopaper. A number of 
studies look at the use of videopapers as a reflective learning tool for prospective 
teachers in different subjects (including mathematics) and its advantages and 
disadvantages over more conventional use of videos, observation tasks and 
assignments (Beardsley et al., 2007; Daniil, 2007; Lazarus & Olivero, 2007), 
adopting very similar methodologies. In these projects, groups of prospective 
teachers were asked to choose one lesson from those they taught in their teaching 
practice and create a videopaper showing self-reflection on their practice, as an 
alternative to the traditional assignment that consisted of writing an essay with the 
same aim. All studies demonstrate the profound insight that is possible when 
teachers use a medium, like videopapers, that allows them to represent and share 
the vitality of their classrooms by means of capturing, preserving, and representing 
events in ways that connect with their world, where different forms of knowledge 
are continually being juxtaposed, as opposed to traditional text-based documents 
(Olivero et al., 2004). Videopapers offer an easy to use tool for teachers to create 
commentaries around teaching episodes, including reflection on their own practice 
and reference to the underpinning theoretical ideas. This provides an essential 
medium for teachers to help them improve their understanding and interpretation 
of their practice (Carraher, Schliemann, & Brizuela, 2000; Derry et al., 2000; Pea, 
2006), leading to knowledge construction and meaning making. 

Research also shows that the process of creating a videopaper is crucial in 
supporting teacher reflection and is qualitatively different from simply watching a 
video from a whole lesson to reflect on one's practice (Daniil, 2007; Lazarus & 
Olivero, 2007). These studies shows that it was the editing process, for example, 
the process of having to select relevant clips to discuss, that fostered prospective 
teachers' reflection, by initiating a more analytical process that might not 
necessarily happen if they had just watched a whole lesson without needing to 
produce a written text. Besides, the multimodal character of videopapers suggests 
that the process of meaning making and knowledge construction, through 
reflection, comes from the two modes (video and text) together. 



Besides the process of creating a videopaper, also the process of reading a 
videopaper has been an object of research. Video has been used in teacher 
education and professional development in different forms since its introduction in 
the 1960s (Sherin, 2004). However, watching a video without text is different from 
reading a videopaper. Reading a videopaper can be described as a dialogue 
between the reader and the author through the text, the video clips and the Play 
buttons that refer the reader to particular clips while reading the text. Smith and 
Krumsvik (2007) present the analysis of the reading processes of and discussion 
around a videopaper integrating key educational theories that prospective teachers 
are normally exposed to. Video illustrations of these theories, taken from the 
authors' own teaching practice, are incorporated in the videopaper. This 
videopaper was read by prospective teachers, teacher educators, and researchers in 
conferences. Smith and Krumsvik argue that this way of using videopapers 
contributes to bringing together communities that would not normally exchange 
ideas around teaching; they also found that the prospective teachers appreciated the 
fact that the practice field was brought to the university and the fact that they could 
"see" the reality of the profession rather than just "hear" about it. 

Overall we can say that videopapers mediate reflection on practice and 
therefore enculturation in the mathematics teaching community for both the creator 
and the reader. The mediation occurs through the multimodal character of 
videopapers, which enables the processes of becoming, doing and experiencing. 
Videopapers crystallise the reflection of the teacher creating them, through video 
and text, and stimulate reflection in the teachers reading them, through the 
dialogue and interaction with the video and the text created by the author. Multiple 
perspectives are elicited, so contributing to the development of professional vision 
and knowledge of mathematics teachers. 

The tools discussed in this subchapter differ from the online learning 
environments previously described in that they are artefacts that are constructed by 
the prospective teachers themselves, with the purpose of representing knowledge 
and eventually communicating and sharing it, as opposed to given artefacts within 
which students may interact. Therefore, what emerges as important is the process 
of writing a blog or creating a videopaper, first as private tools embodying 
personal reflections and representing what is "seen" and "noticed" and then as 
public tools communicating and sharing these reflections with others. The 
collaborative aspect of these tools resides in this double face (creating and 
communicating) and in the interactive process of reading a blog or videopaper, 
through which multiple perspectives are elicited around a shared context, 
represented for example by the video clip in a videopaper. Because these forms of 
discourse are relatively new, they need to be gradually appropriated by both writers 
and readers before they can really become an integral part of teaching and learning 




In the previous subchapters, we have identified key factors that should be 
considered when computer-supported communication tools are introduced in 
mathematics teacher education and that seem to shape the characteristics of online 
interactions, construction of knowledge and creation and support of communities 
of practice. 

The first key factor is the level of structure imposed on the context of 
communication in the community and in the designed learning environment. The 
form in which the activities in online interaction spaces are structured and the 
establishment of clear goals, seem to mediate the process of knowledge building in 
interactive learning environments and the creation and sustainability of 
communities of practice. In some cases, allowing prospective teachers to build a 
space that meets their needs helps the emergence of the sense of belonging to a 
community and supports its sustainability over time (Goos & Benninson, 2004). 
Here, the emergent design of the community contributed to its sustainability by 
allowing the prospective teachers to define their own professional goals and 
values. In this case "communities" emerge since prospective teachers have more 
freedom to share their interests. 

However, in other contexts, it was the pre-determined structure of the activity in 
which prospective teachers were engaged that contributed to the development of 
reflective dialogue, as for example in the learning environments integrating online 
discussions and the analysis of segments of mathematics teaching. A structural 
factor that seems to influence the involvement of prospective teachers in this type 
of learning environment was that the prospective teachers had a clear 
understanding of the goal of the activity (Llinares & Vails, 2007). In this 
intervention in a mathematics methods course, the object of reflection that the 
prospective teachers had to focus on was made salient since the initial phase and 
the conditions were public. In this case, the thematic prompts given to the 
prospective primary teachers seemed to contribute to the development of a shared 
understanding of the situation. Another characteristic that determined the level of 
participation of the prospective teachers was the nature of the questions posed and 
the theoretical information included in the learning environment (thematic 
prompts). The nature of the questions posed in the learning environment 
contributed to keep the messages focused and to generate different interactions that 
favoured the reification of meaning about mathematics teaching as a knowledge 
building process. But a common aspect of all the interactions generated in different 
social interaction spaces is that the activity towards the analysis of mathematics 
teaching or the identification of one-self as belonging to a community of practice 
were mediated by tools, like written text and technological tools (bulletin boards, 
online discussions, writing in a blog, creating a videopaper). Prospective teachers 
indicated that having to write and think about the different issues posed also 
contributed to generating a reciprocal understanding and a sense of belonging to a 
community. One thing that research has begun to point out is that the type of 
participation and the reification processes in these new learning environments 



seem to be dependent on the structure of the activity in which prospective teachers 
are engaged (Zhu, 2006). But the focus on a shared interest helps prospective 
teachers to engage in the joint activities enabling them to share the understanding 
of a given situation (Llinares & Vails, 2007). 

The second key factor to our understanding of prospective teacher learning is 
the role played by new communication tools and by the nature of the generated 
discourse. Communication through forums, bulletin boards, blogs or videopapers 
mediates the process of becoming (learning as identity), doing (learning as 
practice) and experience (learning as meaning) of the prospective mathematics 
teachers. The sociocultural perspective on teacher learning underlines the role 
played by the artefacts built by prospective teachers as communication 
instruments. The prospective teachers' written contributions in an online 
discussion, the process of creating a videopaper or a blog as a way of 
communicating their understanding of facts and situations mediate the process of 
knowledge building and, in some cases, the sense of belonging to a community. 
The important thing is that individuals might come to understand a topic better, 
share resources or have opportunities to define their own professional goals and 
values, when they have to write in order to communicate to others - writing for 
others using the new ways of communication that new technologies provide. 
Writing, as a manifestation of the use of the new available communication 
technologies, is understood from this perspective as a tool for collaborative 
reflection and, at the same time, problem solving. Therefore, "writing" about a 
topic is considered to be a powerful way of knowing. Posting a message in an 
online discussion, exchanging ideas and experiences in bulletin boards, creating a 
videopaper in a reflective context, or creating blogs as ways of sharing knowledge 
and resources through a progressive discourse that takes place in "writing", 
constitute the spaces where prospective teachers' personal interpretations are 
questioned and clarified in an attempt to construct "common knowledge". The 
"text" created within these news ways of communicating functions as an 
improvable object, that provides the focus for progressive discourse and 
simultaneously embodies the progress made; this might allow prospective teachers 
to become acquainted with and understand the topic they are writing about and can 
be seen as a dialogic process of knowledge (Andriessen, Erkens, van de Laank, 
Peters, & Coirier, 2003; Wells, 2002). In these new contexts, discourse enables the 
development and articulation of shared values. 

In summary, the introduction of information and communication technologies in 
teacher education also involves the development of sociocultural perspectives on 
learning that provide new avenues through which teacher educators may attempt to 
understand the process of becoming a mathematics teacher. Despite the scarcity of 
research on the topic, the studies discussed in this chapter have shed light on key 
factors related to the introduction and use of these new tools. Questions have 
emerged too and these call for further research. 




The contribution of S. Llinares was supported by Ministerio de Educacion y 
Ciencia, Direction General de Investigation, Spain, under grant no. SEJ2004- 


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Salvador Llinares 

Departamento de Innovation y Formation Diddctica, 

University of Alicante 


Federica Olivero 
Graduate School of Education, 
University of Bristol 
United Kingdom 




The Role of Technology in Collaboration 

The collaboration of practising teachers in a virtual environment introduces the 
technology tools themselves both as mediators and as participants - ay co-actors - 
in the collaborative process. Although there is a growing literature on the 
collaboration of practising teachers, the role of virtual technology tools is typically 
not addressed In this chapter, we turn our attention to two cases, one in Brazil and 
one in Canada, as we explore how tools mediate and interact in the way teachers 
collaborate and construct knowledge. A challenge in this exploration is that 
technological tools change dramatically over short periods of time. Some aspects 
of teachers ' online learning that are brought to light by the two cases from Brazil 
and Canada are: (I) virtual collaboration can happen in very different ways and 
using very different tools and methods; (2) online technology tools can transform 
abstract mathematics objects like polygons into tangible objects of communal 
attention and action; (3) collaborative knowledge construction tools like wikis help 
re-shape the collaborative process and transform roles played by teachers and 
instructors; and (4) multimodal communication through drawing tools, rich text, 
and video changes the "face " of mathematics. The virtual, non-human objects that 
are part of collaborative collectives of humans-with-media are not tools that we 
simply use for predetermined purposes. Humans-media interactions, which are 
quickly evolving with changes in the online world, are organic, reorganizing and 
restructuring our understanding of what it means for practising mathematics 
teachers to collaborate in a virtual environment. 


Since the mid 1990s, as the WWW became available in the virtual world, there has 
been resurgence and a redefinition of the idea of distance education. This modality 
of education is of course much older and used regular mail and television as the 
main means of communication between students and teacher. The main 
characteristic of this kind of education is that, while ideas were transmitted, 
students and teachers did not share the same space, and until the 1990s, the 
interaction was limited to teacher-to-students dialogue and did not include student- 
to-student dialogue. The Internet introduced the possibility of online collaboration 
among teachers in "distance education" settings, using synchronous (occurring at 
the same time) and asynchronous interaction with different modes of 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 181-206. 
© 2008 Sense Publishers. All rights reserved 


communication such as chat rooms, forums, wikis, videoconferences and 
multimodal ones in which text, pictures, video and voice are combined in different 

Distance education with the strong use of the Internet has been renamed online 
education. This modality of education has been used in undergraduate courses, as 
reported by Engelbrecht and Harding (2005), mainly with the independent learning 
model, which stresses the download of didactical material by learners. In such a 
model, the emphasis is on posting "instructional material on the web". Expressions 
such as "self-learning" and others that negate the role of the teacher are associated 
with this model of online education. 

Interestingly, parallel to the growth of online learning in mathematics teacher 
education and in teacher education in general, there has been a growing interest in 
the collaboration of teachers. Krainer (2003), for example, notes that "Increasingly, 
papers in teacher education refer to some kind of 'communities' among teachers" 
(p. 94). There is also growing evidence that collaboration among teachers is a key 
ingredient for their professional development (e.g., Krainer, 2001; Peter-Koop, 
Santos- Wagner, Breen, & Begg, 2003). In trying to understand teacher professional 
development, many distinctions have been made among terms such as cooperation, 
collaboration, collegiality, teams, networks, and communities to address issues of 
power, conflict, conflict resolution and reflection (Begg, 2003; Krainer, 2003; 
Santos- Wagner, 2003; Lave & Wenger, 1991). Because of the recent availability of 
online collaborative tools, it is not surprising that the role of technology has not 
been addressed in most of the work on teacher collaboration. 

However, some exceptions can be found in the last few years. Literature that 
addresses issues of teacher collaboration in online and face-to-face settings 
identifies a variety of methods for creating a collaborative focus: using multimedia 
cases (McGraw, Lynch, Koc, Budak, & Brown, 2007; Llinares & Olivero, this 
volume), identifying pedagogical issues of common interest (Arbaugh, 2003; 
Groth, 2007), using student work as a focus of reflection and discussion (Kazemi & 
Franke, 2004), and mathematical content (Lachance & Confrey, 2003; Davis & 
Simmt, 2006). A gap in the literature on the collaboration of practising 
mathematics teachers, and the focus of our chapter, is the role of virtual 
environments and tools both as factors mediating teacher collaboration and as co- 
actors in the collaborative process. 

Our work on online teacher collaboration is based on a perspective that 
knowledge is constructed in interactions with others, what has been labelled as a 
Vygostikyan, sociocultural approach. By "others" we also refer to digital tools that 
permeate our new media culture. Borba and Villarreal (2005) see humans-with- 
media as actors in the production of knowledge and they note that humans-with- 
media form a collective where new media also serve to disrupt and reorganize 
human thinking. They base their view in authors such as Levy (1997), who 
suggests that technology itself is an actor in the collaborative process. Levy sees 
technology not simply as a tool used for human intentions, but rather as an integral 
component of the cognitive ecology that forms when humans collaborate in a 
technology immersive environment. 



In our research (Borba & Penteado, 2001; Gracias, 2003; Borba, 2005; Borba & 
Villarreal, 2005; Gadanidis & Namukasa, 2005; Santos, 2006; Gadanidis, 
Namukasa, & Moghaddam, in review) we have used online education models that 
value the interaction among teacher educators and prospective or practising 
teachers by using synchronous and asynchronous interactions. Another 
characteristic of models we used in our courses and of the research that we 
developed involves exploration of what is new in computer technology. The 
medium used is considered in a deep way as a co-actor, that is an active, modifying 
agent that transforms the collaborative process, in the same way that writing is seen 
as being modified by the medium used, be it paper and pencil, or computer word 
processing. Other authors in this area, such as McGraw et al. (2007) and Rey, 
Penalva, and Llinares (2007) do not seem to take into consideration that the effect 
of media is important; they assume writing in online environments to be "neutral", 
as far as media is concerned, in their analysis of the way mathematicians, 
mathematics teachers, prospective educators and practising teachers interact. It is 
too early to try to draw strong conclusions regarding the role of the Internet in 
mathematics teacher education as there is little research on media as a co-actor in 
online education. Therefore, in this chapter, we would like to provide examples to 
inspire those concerned with teacher education to think about this issue which, as 
recently as 2005, was not a major topic in the ICMI study that took place in Brazil 
(, although studies such as 
Pateman, Dougherty, and Zilliox (2003), and Hoines and Fuglestad (2004) began 
to appear at conferences such as the Psychology of Mathematics Education (PME). 
We will first give examples that come from both Brazil 1 and Canada separately and 
then a final one that involved collaboration between both research teams. In these 
examples, we will show that, although we have experienced different kinds of 
online courses and used different types of interfaces, there is a common underlying 
goal of providing tools for interaction and supporting a collaborative culture among 
participants and among teachers and researchers. We will also show how a given 
tool shapes the nature of interaction, stressing, therefore, the role of media in the 
way teachers collaborate and the way knowledge is produced among participants. 


Brazilian mathematics education has organized itself, among other means, in 
research groups. This type of organization has helped us to focus our efforts in 
conducting research. For instance, the first author of this paper participates in 
GPIMEM, a fifteen-year-old research group, registered in our national research 
group database. Since 1993, we have studied the role of different software in 
mathematics education, and how pedagogical approaches that involve students in 
choosing problems to be solved are in resonance with the use of information 

Some of the examples presented in this chapter from the Brazilian side have been presented since 
2004 at PME conferences. See for example Borba (2005); Borba and Zulatto (2006); Borba (2007). 



technology, including the Internet. Since the late 1990s, we have been involved 
with online education, specifically. First, we started studying the scarce literature 
available then in a search for one model of online courses to guide our work. Since 
2000, we offered online courses as a means of experiencing distance education, as 
we continued to study the increasing number of studies and descriptions of online 
distance education. Courses such as these are of paramount importance in Brazil 
due to the size of the country and the concentration of knowledge production in the 
southeastern region, where the states of Sao Paulo and Rio de Janeiro are located. 
Internet- based courses are one way of connecting research centres such as Sao 
Paulo State University (UNESP) with people in remote locations, where the closest 
university may be more than several hundred kilometres away. This kind of 
practice and research illustrate how online education can be a path for social 

To present the way GPIMEM views online education, we will briefly depict the 
way we conceptualise technology in mathematics education as a result of our 
experience with software we used in the classroom. We have developed the 
theoretical notion of humans-with-media (Borba & Vilarreal, 2005) as a means of 
stressing the idea that knowledge is constructed by collectives which involve 
humans and different technologies of intelligence (Levy, 1993), such as orality, 
paper-and-pencil, and information and communication technology (ICT). Different 
humans, or different technologies, result in different kinds of knowledge 
production. Knowledge production involves humans and some medium. This 
notion has provided important insights as we examined how different interfaces, 
such as graphing calculators, function software, or dynamic geometry software, 
affected knowledge production (Borba, 2004). We, as a group, illustrated how 
different media shape the way mathematics is produced. For example, students- 
with-graphing-calculators are more likely to raise conjectures and discuss 
mathematical ideas related to them (see e.g., Borba & Villarreal, 2005) than 
students working without such mathematical modelling and representation tools; 
also, collectives of humans-with-geometry-software-paper-and-pencil have 
integrated simulation and demonstration, as shown by Santos (2006). We were 
curious to know whether the Internet would also be equally as active in the way 
mathematics is done as we have argued that geometry and function software have 

Based on research developed by GPIMEM we discuss the following issues: (1) 
our view about the role of technology in knowing; (2) our model of online courses 
based on interactions; (3) the importance of synchronous relationships; and (4) the 
active role of different internet interfaces on the construction of mathematical 
knowledge by teachers. We elaborate on each of these below and discuss them in 
the context of teacher collaboration. 

Chat as the Main Interface for Online Courses 

In order to learn how to teach online and to develop research, some members of 
our group have been researching how collectives formed of humans-with- Internet 



construct knowledge. For this purpose, among others, we offered several online 
courses for mathematics practising teachers. These courses were based on our 
previous experience with technology, which led us to believe that real-time 
interaction with the teacher is paramount when one works with technology. This is 
why, in most of our courses, we emphasized that at least one part of the course be 
synchronous, either through chat or videoconference. We have offered online 
courses such as Trends in Mathematics Education. These courses have fostered the 
development of communities that discuss issues related to the topics presented: 
teaching and learning of functions and geometry using software, 
ethnomathematics, modelling, adult education in mathematics, critical mathematics 
education, and so on. Eight versions of the "Trends course" have been offered, all 
of them with updated literature discussions along with adopting different virtual 
environments and evolving changes in the model adopted. 

Each course brings together approximately 20 practising teachers online at 
regularly scheduled times over a period of about three months. The first author of 
this chapter and a collaborator taught all the eight courses offered until 2007. 
Almost every week, chat sessions of about three-hours long were scheduled. The 
teachers who took the course each time were, for the most part high school 
teachers, but university level teachers, teacher educators and others, such as 
curriculum developers, also took part in a group that could thus be labelled as 
heterogeneous. It was common in these chat sessions to have simultaneous 
dialogues, since different teachers would pursue different aspects of a given 
problem, or would pose a different problem, or talk about something that happened 
recently in their classroom. Courses were, therefore, designed in such a way that 
interaction was the key word. In 2003 and 2004, we used a free online 
environment, Teleduc 2 , which requires a Linux server, but can be accessed by 
computers that use different platforms. Chat rooms became the principal means of 
synchronous interaction in the course. Preparation for a session would be done 
through asynchronous interactions, mainly e-mail and regular mail. For example, 
prior to a session on ethnomathematics, participants were mailed a book by 
D'Ambrosio (2001), the major proponent of ethnomathematics. All participants 
were expected to have read the book before the session, and two of them would be 
responsible for raising questions to generate discussion. After the class, a third 
teacher would generate a summary for the class that would be published in the 
virtual environment of the course and be made accessible to all participants. 

However, when the objective of the class involved doing mathematics a 
different kind of preparation was required. Problems involving the use of function, 
for example, were sent beforehand to the teachers, and they attempted to solve 
them before the class; during the chat session, different solutions were discussed. 
When we decided to include problem solving sessions, as part of our online course, 
we also started to investigate how mathematics produced via chat might be shaped 

TelEduc is a free plataform for online courses developed by Nied and the "Institute de Computacao da 
Unicamp", chaired by Dr. Heloisa Vieira da Rocha. It is available for download at: 



by this particular medium, and this became an important research question for us as 
mentioned before. 

The problems that we posed to the teachers taking the online courses were 
designed to be solved with the use of plotters such as Winplot, 3 or a geometry 
software, Geometricks (Sadolin, 2000). Prior to a chat meeting, teachers could send 
their solutions to the virtual environment used in a given course, which might then 
be posted, for example, in a tool called portfolio. However, it was not possible to 
simultaneously share a figure with the other course participants. Such a situation 
generated discomfort for some participants. In the 2003 class, prior to a scheduled 
chat meeting with all 20 teachers participating in one of the courses, a problem was 
posed to them regarding Euclidean geometry. Different solutions and questions 
were raised by all participants, but one teacher's reflections caught our special 
attention; during the discussion, Eliane, 4 said: "I confess that, for the first time, I 
felt the need for a face-to- face meeting right away [...] it lacks eye-to-eye contact." 
She then followed up, explaining that discussing geometry made her want to see 
people and to share a common blackboard. In this case, there was no follow-up 
discussion to clarify what she meant. From her comment, we started to think about 
an initial answer to our question regarding the transformation of mathematics in 
online courses: the clash between Euclidean geometry, a symbol of space in our 
culture, and virtual space, a symbol of the beginning of the 21 st Century, may 
permeate things such as doing mathematics. Some of the teachers did feel a need to 
share a screen during the synchronous interactions. Just seeing a solution in a 
portfolio, and commenting on it in another part of the virtual environment was not 
enough. As we will see later in this chapter, there are interfaces in the virtual world 
that can overcome part of this discomfort. For our research, it meant that the 
possibilities of the tool, of the virtual environment, shaped the way teachers were 
producing mathematics collectively. 

A more detailed example will show a different facet of the way chat may 
transform the way mathematics is displayed, and in our view, this signifies changes 
in mathematics. 

In the 2004 class, we posed the following problem 3 to the teachers who 
participated in the course, which is based on a true story that happened in a face-to- 
face classroom, taught by the first author of this chapter: 

Biology students at UNESP, S3o Paulo State University, take an introductory 
course in pre-calculus/calculus. The teacher of this course asks the students to 
explore, using a graphing calculator, what happens when the values 'a', 'b' 
and 'c' in y = ax 2 + bx + c are changed. Students have to report on their 
findings. One of them [Renata] stated: "When b is greater than zero, the 


Eliane Matesco Cristovao, High School teacher, from the 2003 class. 

Translation of this problem and of the excerpt from Portuguese into English was done by the author 
and Anne Kepple. 



increasing part of the parabola will cross the y-axis [...]. When b is less than 
zero, the decreasing part of the parabola will cross the y-axis." What do you 
think of this statement? Justify your response. 

The mathematics involved in the conjecture, and its accuracy according to 
academic mathematics, is developed in detail in Borba and Villarreal (2005). But it 
is interesting to see how the teachers, participants of this online course, dealt with 
it. Some aspects of it were eliminated since they were seen as irrelevant to the 
understanding of the dialogue, or because they were part of a conversation that was 
not associated with the solution of the problem. 

Carlos, a high school teacher, started the debate at 19:49:07 (these numbers 
indicate the hour, minutes and seconds when the message reached the on-line 
course), reporting on what one of his students, in a face-to-face class, had said: 
"When a is negative, or b is positive, the parabola goes more to the right, but when 
a is negative and b is also negative, the parabola goes more to the left." He 
challenged the group to see if the student's sentence could lead them to solve the 
problem. Since the debate was not gaining momentum, the professor of the course, 
the first author of this chapter, tried to bring the group back to what Carlos had 

(19:53:15) Marcelo Borba: The solution that Carlos' student presented 
regarding 'a' and 'b'. Does anyone have an algebraic explanation for it? 

(19:54:53) Tais: It has something to do with the x coordinate of the vertex of 
the parabola. 

(19:55:30) Carlos: after a few attempts (constructing many graphs changing 
the value of 'a', 'b', and 'c') the students concluded that what was proposed 
by Renata [the Biology student who first stated the conjecture in one of 
Borba's face-to face-classes] is really true. 

The issues at stake are distinct. Carlos tried to do what the professor proposed to 
the group, but Tais raised a new issue, the vertex idea. As can be observed on the 
excerpt below, the two issues also have intersections: 

(19:57:07) Tais: Xv = -b / 2a [...] if 'a' and 'b' have different signs, Xv is 

(19:59:16) Norma: I constructed many graphs and I checked that it is correct, 

afterwards I analyzed the coordinates of the parabola vertex Xv = -b / 2a, and 

developed an analysis of the 'b' sign as a function of 'a' being positive or 

negative, then I verified the sign of the vertex crossing. [...] with the 

concavity upwards or downwards, and checked if it was increasing or 

decreasing. [...] did I make myself clear? 


I Norma presented her ideas, which according to the interpretation developed, are 

| similar to the one made by Tais, and can be labelled the vertex solution. After 




further discussion about this, the professor presents another solution based on the 
derivative of y, y' = 2ax + b: 

(20:07:03) Marcelo Borba: Sandra, [...] I just saw it a little differently. I saw 
it [...] I calculated y'(0) = b. [...] and therefore when 'b' is positive the 
parabola will be increasing and analogously [...]. 

Since a few people said they did not understand this comment, the educator 
went back to explain his solution. 

(20:10:59) Marcelo Borba: [...] as I calculate the value of y', y 1 > 0, then the 
function is increasing, and therefore I consider y'(0), which is equivalent to 
the point at which y crosses the y-axis, and y'(0) = b, and therefore 'b' decides 
the whole thing!!!! Got it? 

(20:29:24) Badin: The parabola always intercepts the y-axis at the point 
where the x coordinate is zero. In order for this point to belong to the 
increasing "half of the parabola (a > 0), it should be left of the x v , this means 
x v should be less than zero. Therefore, -b / 2a < is equivalent to -b < 
(remember, a > 0). But -b < is equivalent to b > 0. In other words, if b > 0, 
the point where the graph crosses the y-axis is in the increasing part of the 
parabola. The demonstration for a < is analogous. 

At this point, some of the teachers had been discussing the problem and both 
solutions - the vertex and the derivative - for 40 minutes. The large spaces shown 
by the clock between the different citations from course participants indicate the 
size and number of sections that were eliminated in this paper, as there were about 
four messages per minute. For 10 more minutes, additional refinement and shared 
understanding of the solutions were presented. More examples of people's writing 
about their understanding in the chat are available in the naturally recorded data. 
Educational issues regarding the use of Winplot to explore the problem and 
generate conjectures were discussed. But what is new about the Internet in this 

Before going further, the reader should be aware that some sentences were 
omitted to make it easier to follow the interaction, and that the translation from 
Portuguese into English suppressed most of the informality and typos that normally 
occur in this kind of environment. There were other actors involved in the 
discussion and refinement of the solutions of the problem, but for the purpose of 
clarity, only a few are included here. When we compare the solution presented by 
the teachers - the vertex one - to the original situation that took place in a normal 
classroom situation in 1997, there are similarities and differences. Students used 
graphing calculators to generate many conjectures for the problem relating 
coefficients of parabolas of the type y = ax 2 + bx + c to different graphs. Similarly, 
the teachers used Winplot (or other software, in some examples) to investigate the 
problem just described, and later the problem related to Renata's conjecture. In the 
face-to-face classroom, the professor (author) led the discussion, and eventually 



presented the vertex solution (as he did not know the answer either, at first). The 
students never wrote the explanation for the conjecture. In an on-line learning 
environment based on chat, writing is natural, and everyone involved had to 
express themselves in writing (see also Llinares & Olivero, this volume). Although 
we know that some aspects of writing in a chat situation are different compared to 
writing with paper and pencil, there is a fair amount of research showing the 
benefits of writing for learning (see e.g., Sterret, 1990). However, the data 
presented here is insufficient, and the design of the study is inappropriate, to 
support arguments about "benefits". Still it can be argued that chats transform the 
mathematics that is produced by the participants of the course. The chat tool, 
together with human beings, generate a kind of written mathematics that is 
different from that developed in the face-to-face classroom, where gestures and 
looks form part of the communication as well. We have been building evidence, 
with other examples, of how collectives of humans-with-Internet-software generate 
different kinds of knowledge, which does not mean that the mathematical results 
were different. But if the process is considered, most of us at GPIMEM believe that 
we may be on the way to discovering a qualitatively different medium that, like the 
"click and drag" tool of the dynamic geometry, offers a new way of doing 
mathematics that has the potential to change the mathematics produced, because 
writing in non-mathematical language becomes a part of doing mathematics. At 
this point, it is too early to confirm this, but we believe that this "working 
hypothesis" (Lincoln & Guba, 1985) regarding the transformation of mathematics 
by the Internet is one that we have been pursuing in research developed within 
virtual environments like the ones described, but also in different ones, such as the 
ones we present below. 

In virtual collaboration, the participants that influence the nature and the focus 
of the collaborative process are not only the human participants, but also the 
technological tools and the collaboration affordances and constraints that they 
introduce. In Borba and Penteado (2001), and Gracias (2003), we show that 
"multialogues" - understood as simultaneous dialogues - are a characteristic of 
interaction in chat rooms. Unlike in a regular class, when more than one person 
talks at the same time only when there is group work, in chat, theoretically all can 
"talk" at the same time. In a videoconference environment, new kinds of 
collaboration emerge. Interaction and collaboration, two key words of our model of 
online courses we have, can also be supported drawing on the literature on teacher 
education, where authors such as Hargreaves (1998), Larrain and Hernandez 
(2003), Fiorentini (2004), and Llinares and Krainer (2006) claim that collaboration 
and sharing are powerful actions that generate new knowledge. If we bring these 
ideas to online courses, we have a strong argument for generating courses that 
emphasize interaction not only with the leader of the course, but among 
participants. In a cyclic model, we researched different types of courses offered, 
chose one that emphasizes collaboration, and investigated how different platform 
interfaces for such courses shape the knowledge that is produced, or in other 
words, how different collectives of humans-with-media produce different kinds of 



mathematics and collaboration. As we will see next, different interfaces mean 
different possibilities of online education. 

Video Conference as the Main Interface for Online Courses 

A course, entitled "Geometry with Geometricks", was developed in response to a 
demand from mathematics teachers from a network of schools sponsored by the 
Bradesco Foundation spread throughout all the Brazilian states. The teachers from 
these 40 schools, which include some in the Amazon rainforest, have access to 
different kinds of activities, such as courses that are administered at a pedagogical 
centre located in the greater Sao Paulo area. Following the improvement of Internet 
connections in Brazil, the administrators of the school network realized that online 
courses could become a good option, since sending teachers from different parts of 
Brazil (which is larger in area than the continental US) to a single location to take 
courses was neither cost nor pedagogical ly effective. Lerman and Zehetmeier (this 
volume) point to cost as an important factor for programmes that involve practising 
teachers when goals include encouraging teachers to reflect on their practice 
outside the classroom. The cost factor is related to the size of the country, and the 
pedagogical consideration is related to the fact that teachers would usually 
participate in the courses for a short period of time, with little or no chance of 
implementing new ideas while still taking the course. Our model, based on online 
interaction and applications in their face-to-face classes in middle and high school, 
gained respect gradually within this institution. 

The pedagogical headquarters of this network of schools approached us, asking 
for a course about how to teach geometry using Geometricks, dynamic geometry 
software originally published in Danish and translated into Portuguese. It has most 
of the basic commands of other software such as Cabri II and Geometer Sketchpad, 
and it was designed for plane geometry. As we know from previous research on the 
interaction of information technology and mathematics education, just having a 
piece of software available, and a well-equipped laboratory with 50 
microcomputers, as is the case of these schools, is not enough to guarantee their 
successful use, even if the teachers are paid above average compared to their 
colleagues from other schools. 

In our research group, we designed a course using an exploratory problem 
solving approach similar to that discussed by Schoenfeld (2005); it was divided 
into four themes within geometry (basic activities, similarity, symmetry and 
analytic geometry). Problems usually had more than one way of being solved, and 
they could be incorporated at different grade levels of the curriculum according to 
the degree of requirements for a solution, and according to the preference of the 
teacher. Both intuitive and formal solutions were recognized as being important, 
and the articulation of trial-and-error and geometrical arguments were encouraged. 
We "met" online for two hours on eight Saturday mornings over a period of 
approximately three months. Besides this synchronous activity, there was a fair 
amount of e-mail exchanged during the week for clarification regarding the 
problems proposed and technical issues regarding the software; in addition, also 



pedagogical issues regarding the use of computer software in the classroom were 
raised, for example: should we introduce a concept in the regular classroom and 
then take the students to the laboratory, or the other way around? Pedagogical 
themes were also discussed during the online meetings, in particular in one session 
in which, instead of the students working on problems during the week, they had to 
read a short book about the use of computers in mathematics education (Borba & 
Penteado, 2001). We encouraged teachers to solve problems together in face-to- 
face or online fashion. 

The Bradesco Foundation had already purchased an online platform that allowed 
participants to have access to chat, forum, e-mail, and video conference that 
allowed the download of activities, as well. In our course, participants could 
download problems, and they could also post their solutions if they wanted to, or 
they could send them privately to one of the leaders of the course (the authors of 
this paper). The platform allowed the screen of any of the participants to be shared 
with everyone else. For example, we could start showing a screen of Geometricks 
on our computer, and everyone else could see the dragging that we were 
performing on a given geometrical construction. A special feature, which is 
important for the purpose of this paper, was the capability to "pass the pen" to 
another participant who could then add to what we had done on a Geometricks file. 
In this case technological possibilities transformed the way collaboration could 
happen. Different teachers, who were taking the online course, could lead a 
problem solving activity. As it will be described, there were times when one 
teacher would have the pen, and another would be commenting or giving 
instructions on how to proceed with a given problem. 

This example can illustrate how the convergence of different ideas generates the 
collective construction of knowledge about geometry (content knowledge) about 
use of a given geometry software in the classroom (pedagogical content 
knowledge), and about use of the geometry software itself (technological 
knowledge). This problem involved symmetry. A Geometricks file had already 
been given to them with the figure MNOPQ, presented below (Figure 1). Teachers 
were asked to find the symmetric figure, in relation to axis "q". Teachers were 
reminded in the text that the symmetric figure had to remain symmetric even after 
being dragged. 

Figure I. Picture of the file given to teachers. 



As mentioned earlier, teachers were given each problem before the synchronous 
sessions and could interact with one of the teachers of the course by e-mail or other 
means. This allowed us to sometimes choose issues to start the debate and, at the 
same time, we could also limit undesired exposure of some errors. For this 
problem, the vast majority of solutions used a "count dot approach" in which they 
counted how many dots a given vertex was distant from "q". The result is 
visualized in Figure 2. We invited a volunteer to make a construction. We passed 
the pen to one participant who offered to do so, who in turn was helped by another 
who was acting like a sports narrator. After the construction was done, we asked 
questions such as: is MVWZQ symmetric to MNOPQ? The "argument" was quite 
intense, and the participants were divided. The issue about dragging emerged, and 
we came to the conclusion that if the dragging of a vertex is considered to be 
essential, that solution would not generate a symmetric figure (see Figure 3). 

. . >»/ \p. 

- M ■» 

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Figure 2. The first 
solution of the teachers. 

Figure 3. Dragging and 
passes the test of dragging. 

Different solutions were developed by different actors who took over the pen 
and/or whose voices emerged, including the one presented in Figure 4 - where 
circumferences are used playing the role of a compass in dynamic geometry - that 
would still be symmetric after dragging since, as we pull a point away from the 
axis, for example, the symmetric point will do the same, since it is part of the 
circumference. This issue is relevant, and for authors such as Laborde (1998) and 
Arzarello, Micheletti, Olivero, and Robutti (1998), a straight quadrilateral with 
four equal sides and four angles is only considered a square in a dynamic geometry 
software if it passes the "test of dragging", which means, in this case, that it is still 
a square after different kinds of manipulation are performed. For these authors, if 
the resulting figure is no longer a square, we did not have a construction of a 
square but just a drawing. We agree with these authors that this is an important 
issue, even though in our course, we also emphasized the relevant role of "drawing 
solutions" depending on the grade level, the complexity of the problem, or as a 



path for more complex solutions. Other examples such as these were developed 
during this course. Teachers were overwhelmingly positive about the idea of 
collective problem solving during the synchronous videoconference sessions, 
especially when they compared the experience with others in which just we, the 
leaders, would present our solution or present one of their solutions in a more 
expository manner of teaching. Our own assessment, as leaders, was that this 
virtual collaboration created a bonding among teachers that was not experienced 
when we did not use this technical feature. The issue about drawing versus 
construction is not new in the literature; what we believe to be original in the 
episode reported is the fact that it came from an online course, and that it resulted 
from collaboration among teachers. In such collaboration, they learned from each 
other, which is considered important by authors such as Lin and Ponte (this 

: : :, ^xs 

/'./.\. . .y. . ./..-.-■ 

Figure 4. The second solution. 

Hargreaves (2001) proposed that teaching is a paradoxical profession. On the 
one hand, everyone expects from teachers, even in developing Third World 
countries, efforts to help the construction of learning communities in the new 
"knowledge society". Even though most teachers were not educated in a time when 
microcomputers were extensively available, they are required to teach using 
computers and software in the classroom. On the other hand, there is discourse in 
governments and segments of society that claims we should have a "shrinking" of 
the state due to the needs of this knowledge society, and teachers become one of 
their first victims with funding cuts for sectors such as education. Teachers should 
enhance the knowledge society, even though they are some of their first victims. 
Whether teachers are aware of this dilemma or not, they feel the need, and there is 
institutional pressure, to constantly be working on their professional development. 
Integrating software into the face-to-face classroom is still a challenge as teachers 
enter a risk zone (Penteado, 2001) in which the students sometimes know better 
how to manipulate computers and often come up with original solutions to a given 



problem, or even create new problems which are not easy to handle. Soon the 
demand to incorporate the Internet in the classroom should rise as it seems like we 
are moving towards a "blended" learning approach, in which virtual and face-to- 
face interaction will happen "in" the classroom. 

It makes sense, therefore, that teachers experience online courses, in particular if 
one considers the cost factor, as in the case described above. Reimann (2005), in 
his plenary at PME 29, emphasized the need to build artefacts on the Internet that 
could foster collaboration. We agree and believe that we are helping to construct 
collaborative practices within online continuing education courses offered to 
mathematics teachers. This type of collaboration was possible due to a design in 
the platform that allowed us to do so. Of course, having the possibility is just one 
step, and a pedagogical approach that enhanced participation should always be 
pursued. But we want to emphasize that we should also be creating demands for 
technical developments in online platforms, and this is an area in which our 
research group, GPIMEM, has started working in the last few years. 

We would like to say that once design features are incorporated either into 
software, such as dragging, or into online platforms, like the "pass the pen" option, 
participants become co-actors in the process of creating and recreating knowledge. 
Different media, different people, imply different perspectives in the knowledge 
constructed. That is why we believe that using the construct of humans-with-media 
is useful to analyse educational practices that use technology, since they foster the 
search for specific uses of an available interface. "Pass the pen" was a unique 
characteristic of this platform that made possible the co-construction of 
mathematical knowledge in an online course. 

Engelbrecht and Harding (2005) have presented a study that shows that many 
courses offered are based on independent learning with very little interaction 
among participants. Those who take courses like these are expected to download 
material and learn by themselves, and they are then assessed through some kind of 
standardized test. Many teachers can be enrolled in courses like these, thus 
generating profit for the organizers, and they dismiss the role of interaction in 
teachers' professional development. At the other end of the spectrum are courses 
that use the Internet as a means of generating quick feedback among participants. 
For example, forum is used for asynchronous relations so that participants express 
their ideas and doubts and post solutions to problems. Synchronous relationships 
that employ chats, videoconference, and other tools, make it possible to share ideas 
in real time, even if people do not share the same space. Courses such as these are 
based on the idea that media should not be domesticated, which means that we, as 
mathematics educators, should try to design curricula that fully explore the 
possibilities of new media or interfaces. In the examples presented, we emphasized 
how mathematics gains different characteristics, and teachers collaborate in 
different ways, depending on the interface used. At GPIMEM, we are in the 
process of learning from these experiences to shed light on face-to-face 
mathematics education, as we are more knowledgeable about the role of different 
media in the production of knowledge. As the reader will see, there are 
commonalities and particularities as we learn some from the Canadian experience. 




Our experience, in the Canadian case, based on online education courses offered to 
practising teachers at the Faculty of Education, University of Western Ontario, in 
Canada, points to the following issues: (1) online courses versus face-to-face 
courses; (2) asynchronous versus synchronous online course communication; (3) 
text-based versus multimodal communication; and (4) read-only versus read/write 
communication. We elaborate on each of these below and discuss them in the 
context of teacher collaboration. 


Our Faculty of Education has three teacher education programmes: a teacher 
education programme, a continuing teacher education programme, and a graduate 
programme (masters and PhD). Our continuing teacher education programme 
offers over 150 online courses to Ontario teachers, with approximately 5,000 
teachers taking our courses annually. To support the development of community 
and collaboration, the number of participants in each of the courses we offer is 
capped at 25 teachers. These courses are part of a provincially mandated and 
certified regimen of additional qualifications which lead to teacher professional 
development and in some cases to salary increases. Six of these courses are 
specifically for mathematics teachers. In addition, we offer three mathematics-for- 
teachers courses which are not part of the provincial regimen of courses. Our 
graduate programme offers a fully certified online Master of Education degree in 
addition to its traditional face-to-face masters and PhD degree programmes (where 
some courses are also offered either fully or partially online). Our prospective 
teacher education programme offers some components of its mathematics and 
language arts programme online, where we replaced large group lectures with 
online content and discussion (Gadanidis & Rich, 2003). 

Face-to-Face Versus Online Classrooms 

In the case of courses for practising teachers, our experience indicates that most of 
them prefer online as opposed to face-to-face courses. Over ten years ago, our 
continuing teacher education programme was fully face-to-face, and we offered our 
courses in the evenings, at weekends and during the summer break. We offered 
courses on campus as well as in remote areas, with approximately 2,700 teachers 
taking our courses annually. However, once we started offering courses online, 
teachers opted for these rather than the face-to-race courses. Currently, our 
continuing teacher education programme is approximately 95 percent online. It is 
important to note that we do still offer the face-to-face courses, however teachers 
choose not to take them. It is also important to note that there are other providers 
offering the same courses, both face-to-face and online, and they have experienced 
a similar trend. The main reason for this shift from face-to-face to online, based on 



surveys we conduct on a regular basis, is that teachers lead busy lives, in and out of 
school, and having to attend classes at set times is a scheduling burden. 

Asynchronous Versus Synchronous Classrooms 

Teachers also tell us that the asynchronous nature of our online program makes it 
appealing. Rather than having to schedule their lives around an arbitrarily 
scheduled class, they can choose to participate during a time in the day or night 
that is most convenient for them. The indication we have from teacher feedback is 
that if our online courses used a synchronous mode, many teachers would either 
opt to take online courses from another provider that used an asynchronous mode 
or they would take a face-to-face course. It is important to note that we have a 
significant number of Ontario teachers that teach overseas in countries such as the 
Middle East and Singapore and it would not be possible for them to participate in 
synchronous discussions because of time zone differences. The fact that most of 
our practising teachers choose our online rather than our face-to-face courses, and 
indicate that they prefer asynchronous rather than synchronous online discussion 
environments, does not necessarily mean that one mode is educationally better than 
another, or that it is a simple either-or choice between modes (as hybrid 
environments can and do exist). However, it is interesting to consider some of the 
differences between these modes and how the differences may impact on teacher 
online collaboration. 

Face-to-face and synchronous online discussions are temporal experiences. 
They occur in real time and they have a linear quality. In the Brazilian case we 
noted that online synchronous chat is different to face-to-face synchronous 
communication because the former allows for more than one discussion theme to 
be conducted simultaneously, with chat postings woven into a complex tapestry of 
ideas being discussed simultaneously. This would be analogous to the (impossible) 
face-to-face situation where teachers work in small groups but somehow everyone 
can hear what everyone else is saying without the dialogues overlapping. The 
tapestry of multi-theme postings in an online synchronous chat may cause some 
confusion or disorientation. However, as we noted in the Brazilian case, this multi- 
linear, multi-tasking environment can provide a novel, complex and rich 
experience of collaboration. 

In the Ontario case, a chat tool does exist in the e-learning platform we employ. 
However, the chat tool is rarely used. The asynchronous online discussion used in 
our courses introduces some interesting affordances for collaboration. Unlike 
typical synchronous discussions (especially when the synchronous communication 
is oral rather than textual), where it is possible for a small number of people to 
dominate the discussion, asynchronous discussion makes it possible for everyone 
to contribute to every discussion theme. In fact, this is an expectation in our on|ine 
courses. A significant part of the assessment (typically about 30%) focuses on the 
discussion component, and teacher participation is assessed based on its frequency, 
regularity and quality. Another difference is that in an asynchronous environment, 



teachers can take more time to think about the ideas of others and to craft their own 
responses before posting in the online discussion. 

Textual Versus Multimodal Communication 

Until the end of 2004, our online courses used a platform where communication 
was primarily textual. However, in 2004, we made a decision to offer some of our 
mathematics course for elementary teachers in an online setting. This created a 
challenge because of the lack of an ability to communicate visual aspects of 
mathematics, like diagrams. Consequently, late in 2004, we developed a new 
online platform (Gadanidis, 2007) that offered multimodal communication by 
allowing users to embed the following within discussion postings: rich text (using a 
text editor that is similar to those found in a word processor); diagrams (using a 
built-in draw tool); video or audio (using a built-in tool that allowed the capture of 
video or audio from a web cam); as well as multimedia content, such as JPEG 
images and Flash interactive content. The immediate difference this made to online 
discussion was the visual appeal. Opening a discussion thread the reader was faced 
with postings where text was bolded, coloured and formatted, and accompanied 
with colourful mathematical drawings and images. Figure 5 shows a drawing 
created by an elementary teacher to illustrate her conception of what parallel lines 
may be transformed onto a sphere, a flat piece of paper and a rolled piece of paper. 


Figure 5. An elementary teacher uses the Draw Tool to show 
three representations of "parallel" lines. 

Figure 6 shows the diagram an elementary teacher created to explain to another 
teacher how slope is calculated. Does this type of communication make a 
difference? Kress (2003) and Kress and van Leeuwen (1996, p. Ill) suggest that in 
a digital environment "meaning is made in many different ways, always, in the 
many different modes and media which are co-present in a communicational 
ensemble". In terms of mathematics meaning making, being able to communicate 
via images and user-created diagrams adds layers of meaning and elaboration that 
enhance how ideas are expressed and understood. In addition, the ability to add 
video in the discussion postings (using a web cam) makes the online discussion 



feel personal. For example, prior to a face-to-face graduate seminar in Brazil, video 
postings were used to introduce the Canadian instructors and the Brazilian students 
to one another and to introduce ideas to be discussed in the seminar. Also, the 
ability to add video allows for embodied communication of mathematics, through 
gestures and the use of physical materials. Figure 7 shows how elementary school 
students in Brazil used the video capture tool to illustrate how L patterns can be 
used to physically represent odd numbers and their sums as square patterns. With 
the steady growth of bandwidth, the mode of online interaction and the content 
generated are increasingly multimodal. It makes sense that online discussion and 
collaboration among mathematics teachers would use the multimodal 
communication tools available. 

ender o que eu quero 



Figure 6. An elementary teacher 's 
explanation of slope. 

Figure 7. Students in Brazil posting 
videos of physical patterns. 

It should be noted that because the "Mathematics-for-Teachers" courses 
developed in the Ontario case were aimed at elementary teachers, the drawing tool 
was sufficient for most needs to communicate visual aspects of mathematical ideas. 
If this platform was used for discussing more complex mathematics, this could be 
accommodated by embedding Flash applets in discussion postings that allow 
teachers to explore more complex ideas using graph plotters, probability 
simulations, etc. 



Read-Only Versus Read/Write Communication 

The Web is today in the process of transforming itself from a read-only to a 
read/write environment. In terms of online education, this transformation is 
perhaps best reflected through the use of wilds. A wiki is a collaborative website 
that can be edited by anyone who has access to it. When in late 2004 we created a 
platform that allowed for the multimodal communication discussed above, we also 
created the ability to use wiki discussion postings. When someone creates a 
discussion posting, they have the option of making it a wiki posting. This means 
that everyone else in the discussion can edit that posting. 

The wiki feature has been used in a graduate course called the "Analysis of 
Teaching" to create collaborative tasks that focus on mathematics pedagogy. For 
example, we have used the transcript of a typical grade 8 US mathematics lesson as 
depicted in the 1995 TIMSS Video Study (see excerpt below), by posting this 
transcript in a wiki posting and then asking teachers to collaboratively edit the 
transcript in ways that would improve the quality of the lesson. This is an 
interesting and rather complex collaborative writing experience in that as teachers 
start editing the transcript, other teachers have to adapt their own ideas to fit with 
the edits made up to that point. This does not mean that they cannot edit the edits 
of other students, but rather that they have to keep the changing flow of the lesson 
in mind as they make edits: they cannot simply make an edit to one section without 
taking into account the edits that precede and follow. That is, students need to 
understand the edits of other students - seeing the lesson through the pedagogical 
lenses of others - and then negotiate a synthesis of their edits with existing edits. 
Thus the final edited lesson is the result of sophisticated collaborative editing 

Teacher: Here we have vertical angles and supplementary angles. [...]. Angle 

A is vertical to which angle? 

Students: 70. 

Teacher: Therefore, angle A must be [...]. 

Students: 70. 

Teacher: 70, right. Go from there. Now you have supplementary angles. 

Don't you? [...]. Now, what angle is supplementary to angle B, I mean, I'm 

sorry, Angle A? 

Students: B. 

Teacher: [...] and so is [...]? 

Students: C. 

Teacher. Supplementary angles add up to what number? 

Student: 180. 

The TIMSS Video Study (Stiegler, 1999) activity is one of the first collaborative 
writing activities we use in the "Analysis-of-Teaching" course, to get teachers 
comfortable with writing collaboratively and editing the ideas of others. Because 
they are editing a transcript that is not their own, it is easier for them to make edits 
without feeling that they might hurt someone's feelings. Later on in the course, 



students post transcripts from their own teaching, and discuss these in wiki 
postings. Our experience at the graduate level is that teachers are very hesitant to 
edit the ideas of others or to have their own ideas edited. One other thing that we 
do to help shift their perspective is to discuss how the peer review process works 
for scholarly writing. For example, when we submit a paper for possible 
publication in a scholarly journal, we receive feedback from the reviewers about 
edits that we could make to improve the paper. Sometimes, these edits are written 
on a separate piece of paper, sometimes they are written in the margins, and 
sometimes there are suggested edits in the text of the paper. This is typically a 
tremendous learning process, both for the reviewer and for the author. We thus try 
to emphasise that peer review and peer editing is a natural and rewarding scholarly 
experience. Nonetheless, some students never fully engage in the process, and 
instead make superficial edits or react negatively to edits of their own work. It 
takes more than a couple of collaborative writing activities to shift some students' 
view of graduate work as personal and private and of the instructor as the only 
person who has the right to give feedback and make editing suggestions. 


Borba and Villarreal (2005) have argued how different software interfaces interact 
with our cognition and reorganize our thinking. Our thinking about online 
education has been disrupted and reorganized as we have used and thought with 
emerging online affordances like multimodal and read/write communication. For 
example, using a wiki in our online teaching is a very different experience than 
teaching in a physical classroom. It is also very different from using simple text- 
based discussion platforms with read-only discussion postings. Using a wiki does 
not only disrupt and reorganize our thinking about how we structure classroom 
interaction, it also becomes a lens that changes how we see other aspects of our 
online teaching, such as course content, evaluation practices, our role as 
instructors, and generally what constitutes knowledge and how it is or should be 
constructed collaboratively in an online environment. The emerging multimodal 
and collaborative tools of online teaching and learning environments are not simply 
tools that we use for predetermined purposes. Rather, they can be seen as co-actors 
in the cognitive ecology of online environments, existing in a complex, organic 
relationship between humans and technology (Gadanidis & Borba, 2008). 

Collaborating without Geographical Boundaries 

From the examples presented in the Brazilian cases and in the Canadian cases, the 
reader may be led to believe that online education is a solution for many of the 
problems of teaching education, and that it brings no problems. In the Canadian 
case, for example, where teachers have the option of choosing an online or a face- 
to- face version of a course, they overwhelmingly opt for the online version. In the 
Brazilian case there has been very positive assessment of the online courses made 
not only by the teachers who participated in the courses, but also by administrators 



of the private schools who paid for the courses attended by their teachers, as 
reported by Borba (2007). On the other hand, in other courses, in which there was a 
fair number of teachers dropping out, members of the research group based in 
Brazil attempted to know the reasons why teachers left the course, but they could 
not, since there were not enough respondents. In addition, in the Brazilian case, a 
few problems have already been identified, such as the lack of a common 
geometric figure to share synchronously in some environments, but at the same 
time we have also shown how different interface such as videoconferencing can 
solve such a limitation. But in other papers we have pointed out limitations such as 
technical difficulties to deal with a given platform or even inability to type fast 
enough in order to participate in a session (Borba, Malheiros, & Zulatto, 2007). 
Other authors such as Ponte and Santos (2005) and Ponte, Oliveira, Varandas, 
Oliveira, and Fonseca (2007) have shown how different teachers live this 
experience in different ways. They report that although some teachers are positive 
about the online experience, there are others who are far from that, for reasons 
which include the fact that once you "say/write" something it is recorded 
electronically and accessible for everyone to read. So we can say that the modality 
of online courses for teachers is still open for debate and for research. Moreover, 
we need to investigate whether styles of learning are connected to the medium 
used. It may be the case that some teachers prefer online and others like face-to- 

One aspect of teachers' online learning that is brought to light by the two cases 
from Brazil and Canada is that collaboration can happen in very different ways and 
using very different tools and methods. For example, if you read only the Brazilian 
case, you might appreciate the value of synchronous communication as a tool for 
online collaboration. On the other hand, if you read only the Canadian case, you 
might appreciate the value of asynchronous communication as a tool for online 
collaboration. Another aspect that is brought to light by both cases is that the online 
world is changing at a rapid pace. In both cases, the technological tools used 
changed dramatically over short periods of time. The reason for this is the rapid 
development of new online technologies coupled with the rapid growth of Internet 
access and bandwidth. The text-based, read-only online world of a few years ago is 
rapidly evolving into a multimodal, read/write social networking environment 
(Sprague, Maddux, Ferdig, & Albion, 2007). This is bound to have an impact on 
the virtual collaboration of practising mathematics teachers. The question is how 
widespread will this impact be? In a review of online education, Sprague et al. 
(2007, p. 158) suggest "that so-called 'early-adopters' of technology may have 
made up the majority of faculty and students who have so far been involved in the 
online education phenomenon". It will be interesting to look back five or ten years 
from now and see whether the use of the new collaborative affordances of the 
WWW is also limited mostly to the "early-adopters" or whether their use pervades 
online mathematics teacher education. In the same direction, it should be 
interesting to confirm whether, in fact, the online world has been "an actor" in 
transforming mathematics. This has been proposed by Borba and Villarreal (2005) 



in the case of the introduction of given software, and in the way we are proposing 
for online mathematics teacher education. 

Given the ongoing development of the WWW as a social/collaborative 
environment, we believe that a promising path for research is the one that 
investigates the synergy between the virtual tools used and the collaborative nature 
of the online education/professional development of practising mathematics 
teachers. Moreover, the virtual tools that are part of collaborative collectives of 
humans-with-media are not tools that we simply use for predetermined purposes: 
they are active "participants" in the collaborative process. Human-media 
interactions, which are quickly evolving with changes in the online world, are 
organic, reorganizing and restructuring our understanding of what it means for 
practising mathematics teachers to collaborate in a virtual environment. For 
example, we have seen in the Brazilian case how online technology tools can 
transform abstract mathematics objects like polygons into tangible objects of 
communal attention and action. What changes mathematically and what changes in 
terms of the collaborative process when mathematical objects become communal 
objects? We have also seen in the Canadian case how multimodal communication 
through drawing tools, rich text, and video changes the "face" of mathematics. 
How does mathematics change when it is expressed in multimodal forms? How 
does a media-rich environment affect the collaborative process? 

In typical online teacher education settings, the mode of communication is 
textual. Texts on chats and forums are changing the way mathematics is expressed. 
Bringing everyday language to the forefront of mathematics communication is 
already transforming mathematics that is constructed in online mathematics 
education. In the cases presented, communication also involved diagrams, pictures, 
video and interactive content. These modes of communication are increasingly 
changing the mathematics that is constructed in such online environments. We 
believe that these are examples of how non-human authors become co-actors in the 
production of mathematical knowledge. 

In the Brazilian case we can see the "passing the pen" tool transforming 
collaboration as teachers can collectively and synchronously solve a problem in a 
natural way. In the Canadian case, we can see teachers learning to have communal 
production, which can be also seen as being shaped by the possibilities that the 
wiki-like environment they used for asynchronous communication. We believe that 
the examples presented in this chapter illustrate how tools become co-actors in the 
way teachers collaborate and construct knowledge. Design of online tools is a topic 
we should be interested as these non-human objects can become parts of 
collectives of humans-with-media that produce knowledge. To study the role of 
online tools on the role of collaboration in teacher education seems to be a 
promising path for research, as the examples presented are not sufficient for us to 
assure that online tools shape mathematics and collaboration. At this point, it 
seems that we have to see virtual tools as co-actors in the collaboration of 
practising teachers as a working hypothesis to be confirmed, rejected or re- 
elaborated in the future. 



Also, since the social nature of the WWW appears to be permeating our society 
(with the emergence of social networking sites such as MySpace and YouTube), it 
might be worth exploring less formal or perhaps emergent collaborative 
relationships among mathematics teachers. It might also be worth exploring the 
effect of the pervasive nature of WWW on equity of access, as both the Brazilian 
and the Canadian cases involve both urban as well as geographically remote areas. 


Although they are not responsible for the content of this chapter, the authors would 
like to thank Ricardo Scucuglia, Marcus Maltempi, Silvana Santos, Sandra 
Barbosa, Ana Paula Malheiros, Regina Franchi and Rubia Zulatto, members of 
GPIMEM, UNESP, for their comments on earlier versions of this chapter. We 
would also like to thank CNPq, FAPESP and SSHRC, funding agencies of the 
Brazilian and Canadian government for partially funding research presented in this 
chapter. Finally we would like to thank the editors of this book for the review of 
this chapter. 


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Marcelo C. Borba 

Department of Mathematics 

Graduate Program in Mathematics Education 

Sao Paulo State University 


George Gadanidis 
Faculty of Education 
University of Western Ontario 








Towards Multidirectional Analyses of Learning across Contexts 

This chapter focuses on supporting the teaching and learning of mathematics 
through school development. School development entails organizational learning 
and the use of social, structural and material resources to support teaching and 
learning: how are schools or mathematics departments organized to support the 
teaching and learning of mathematics? I begin by reviewing why and how 
researchers have conceptualized the school as a productive setting for supporting 
and improving the teaching and learning of mathematics. I draw on literature from 
organizational learning and professional communities to show how recent studies 
theorize improving instruction and student learning. Much of the literature 
emphasizes mechanisms for such improvement to be highly dependent on how 
schools employ resources and how they are organized to support teachers' 
collective learning together outside of the classroom. The chapter ends by 
proposing new directions for research on professional inquiry and its relation to 
the improvement of teaching and learning mathematics and school development. 


In this chapter, I take up the issue of how schools cultivate organizational supports 
for teaching that aims to successfully engage all kinds of learners in complex 
mathematical learning. In particular, I hold a view of mathematical learning that 
respects students' intellectual integrity and considers schools to be places for 
vibrant intellectual engagement in which students and teachers develop 
sophisticated understanding of mathematics, identities as mathematical thinkers, 
and are engaged in ongoing inquiry. My interest in school development is to 
understand what allows the school community to continually strive to improve 
mathematics instruction in order to strengthen student learning. Much of the 
literature on school development naturally focuses on supporting teacher learning 
and building teacher capacity (see e.g., Borko, Wolf, Simone, & Uchiyama, 2003; 
Lerman & Zehetmeier, this volume; Nickerson, this volume). Thus, a large portion 
of this chapter is devoted to research focused on cultivating professional 
communities where teachers are the major players. In considering school 
development, 1 view membership in a school's professional community to include 
teachers, instructional leaders (e.g., coaches), and administrative leaders. A school 

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 209-230. 
© 2008 Sense Publishers. All rights reserved 


development perspective emphasizes the interdependence of professionals in the 
school community - how schools carve out time and space and employ social, 
intellectual, and material resources to support joint inquiry among teachers, 
instructional leaders and administrators (Collinson, in press; Lam pert, Boerst, & 
Graziani, under review). I review how researchers have conceptualized and 
described this joint inquiry and propose directions for future research. 


What Is Organizational Learning? 

One of the strengths of the mathematics education literature has been our careful 
attention to supporting teachers to improve their instruction. My goal in this 
chapter is to harness our understandings of supporting individual teachers to think 
about school-wide professional learning (see also Knapp, 1997; Krainer, 2006; 
Oliveira & Hannula, this volume; Perrin-Glorian, DeBlois, & Robert, this volume). 
One clear area of growth for mathematics education research on school 
development is to develop a framework for explaining how activities engaged by 
teachers and other professionals at the school level achieve organizational purposes 
and affect student learning outcomes (Boreham & Morgan, 2004). Collinson and 
Cook (2007) define organizational learning as "the deliberate use of individual, 
group, and system learning to embed new thinking and practices that continuously 
renew and transform the organization in ways that support shared aims" (p. 8). 
According to Boreham and Morgan (2004), when organizations learn, "co-workers 
transcend the boundaries which separate them from colleagues, establish a 
common (and expanded) understanding of the object of their joint activity and 
make a collective decision on how to achieve it" (p. 313). In recent work, Lampert 
et al. (under review, p. 35) have been examining how organizational assets - 
material, intellectual, and social - become resources to support successful teaching 
of complex learning; they explain this view in the following way: 

Assets are effective in supporting the common practice of ambitious 
instruction when they are widely used as tools for solving common 
instructional problems. When common use is enabled by the organization of 
practice, assets can shape the culture of teaching and learning. For this 
reason, our analysis has focused not only on the nature of material, social, 
and intellectual assets, but also on how instruction at the school is organized 
so that assets can become resources that are routinely drawn upon to support 
a challenging approach to teaching and learning across the school. 

Lampert et al.'s views echo other researchers who suggest that we need to move 
beyond making direct causal links between resources available at a school level 
(such as adequate funding, leadership, highly qualified teachers, curricular 
materials, release time for teachers) to student achievement. Instead, the argument 
is to examine how such resources are used to support instruction, which mediates 
student achievement (see also Cohen, Raudenbush, & Ball, 2003). 



Professional Inquiry as a Key Resource in Supporting School Development 

The existence of robust professional inquiry at the school level has received a 
considerable amount of attention as one of the key resources in supporting school 
development. Promoting collective professional inquiry within a school emerged in 
part as a response to a large body of organizational and classroom research that 
documented the successes and failures of teachers' attempts at implementing 
reform practices designed to change classroom instruction. Changing daily routines 
inside classrooms, the goals and purposes of mathematics instruction, and 
dominant discourse patterns and relationships between teachers and students have 
proven difficult. Significantly impacting classroom instruction is challenging 
because of what organizational researchers have labelled the loose coupling 
between traditional school structures, management, and teachers' everyday 
decision-making. While teachers may enjoy considerable autonomy, this loose 
coupling in traditional organizational structures creates barriers for instructional 
reform (Weick, 1976). More recent work on teachers' responses to reform 
initiatives suggests that the congruence, intensity, pervasiveness and voluntariness 
of the reforms themselves have an effect on how teachers change their classroom 
practices (Cobum, 2004). Decoupling may be one of many responses teachers have 
to reform initiatives. Others include rejecting reform visions, assimilating, 
accommodating, or developing parallel structures to reform practices. 

To counter forces that keep teachers isolated and encourage idiosyncratic 
responses to instructional initiatives, researchers have attempted to build models 
that are built into the regular workday, arguing they have more promise of being 
maintained, sustained and integrated into teachers' practice than an occasional 
weekend or summer institute (see also Jaworski, this volume; Lerman & 
Zehetmeier, this volume). A focus on schools as a unit of change means that a 
collective intellectual culture needs to be cultivated among key professionals 
(administrators, instructional leaders, and teachers) who work within the 
organization. Basing professional inquiry within the workplace can allow school 
professionals to work together to make sense of demanding mathematics teaching 
and coordinate their efforts across grade levels. Importantly, collective inquiry 
serves to break down the dominant "egg crate modef of schooling - teachers 
housed together in a school yet working individually in their own classrooms with 
little coordination and collaboration with colleagues in the same building. The 
dominant cellular model has promoted isolation and norms of privacy (Lortie, 
1 975). That said, it is important to note, as many researchers have pointed out, that 
simply bringing teachers and leaders together in regular meetings does not mean 
they will be able to critique and learn from their practice. How learning 
opportunities are structured and enacted make all the difference in teacher learning. 

To the extent that we have conducted cross-national comparisons of teaching, 
we know that variation does exist in what counts as "doing" school mathematics. 
The case could be made that a view of mathematics as a static body of procedures 
to memorize and apply with fidelity is a vision that many mathematics educators in 
the US try to move away from in their professional development work with 



teachers. Much of the literature about supporting instructional improvement is 
aimed toward helping teachers learn to treat mathematics as a sense-making 
enterprise that involves argumentation, reasoning, and justification at its core (see 
also Seago, this volume). Cross-national comparisons have helped us see that 
expository teaching approaches are not necessarily universal. In elementary 
classrooms in Japan, for example, significant time is spent on justifying and 
connecting mathematical ideas (Hiebert, Stigler, Jacobs, Giwin, & Gamier, 2005). 
It may be fair to say that a procedural approach to teaching and learning 
mathematics is nonetheless ubiquitous (Giwin, Hiebert, Jacobs, Hollingsworth, & 
Galli more, 2005). Although some variation exists internationally in cultural scripts 
for teaching (Clarke, Emanuelsson, Jablonka, & Mok, 2006; Clarke, Keitel, & 
Shimizu, 2006), static notions of learners, especially durable notions of ability and 
competence may be more pervasive across the globe. Schooling institutions are 
typically organized to classify, sort, and promote students with higher 
mathematical ability and higher achievement (e.g., Spade, Columba, & Vanfossen, 
1997). Schooling practices can embody a view of ability that often takes 
intelligence as an inherent and stable trait (Boaler, 2002; Horn, 2007). Reforming 
mathematics instruction at a school level is often coupled with trying to upend 
static views of students' ability and intelligence in order to subvert tracking, 
grouping, and classification systems that keep students in ability bins (e.g., Boaler, 
2002; Horn, 2007; Oakes, 1985; Rousseau, 2004; Sfard & Prusak, 2005). This view 
sees schools and classrooms as "necessarily cultural and social spaces that can 
perpetuate social inequities by positioning multiple forms of learning and knowing 
as 'having clout'" (DIME, 2007, p. 407). Promoting school development has the 
potential to enable school professionals to interrogate how they view students in 
relation to instruction and how their actions and policies privilege certain students 
over others. 

Visions of Ambitious Teaching, Teacher Learning and the Meaning of Inquiry 

Current research on teaching and learning puts forth a vision of teaching 
mathematics that has been labelled a number of different ways: teaching for 
understanding, reform-oriented teaching, standards-based instruction, problem- 
centred instruction, inquiry-oriented teaching. These terms have been used to 
convey both a level of intellectual rigor, and the nature of the classroom 
community needed to achieve that rigor. Lampert and colleagues (under review) 
use the term ambitious teaching to mean, "adjusting teaching to what particular 
students are able to do (or not)" (p. 2) in order to engage all kinds of students in 
complex problem solving activity, mathematical reasoning, and justification; they 
identify the challenges of ambitious teaching to include (p. 35): 

- Teachers need to move around flexibly in the multiple dimensions of subject 
matter in relation to student performance, adjusting teaching to learning. 

- Teachers need to create an environment where students are willing and 
motivated to take the risks that intellectual performance entails. 



- Teachers need to take what students can do as an integrated indication of what 
they know and what they need to learn rather than breaking subject matter into 
meaningless bits of information. 

To meet such challenges, it seems reasonable to argue that teachers must have 
access to substantive learning opportunities for themselves. The literature on 
teacher learning is replete with uses of the term inquiry. Richardson (1994) 
distinguishes formal educational research from teachers' own practical inquiry by 
describing how practical inquiry aims to help teachers change their instructional 
practice or increase understanding by studying their own contexts, practices, or 
students. When conducted with colleagues, practical inquiry can develop a local 
sense of shared norms and local standards of practice. Cochran-Smith and Lytle 
(1999, p. 288) describe inquiry as stance: 

Teachers and student teachers who take an inquiry stance work within inquiry 
communities generate local knowledge, envision and theorize their practice, 
and interpret and interrogate the theory and research of others. Fundamental 
to this notion is the idea that the work of inquiry communities is both social 
and political; that is, it involves making problematic the current arrangements 
of schooling; the ways knowledge is constructed, evaluated, and used; and 
teachers' individual and collective roles in bringing about change. 

The critical perspective reflective in Cochran-Smith and Lytle's view of inquiry 
as stance is reflected in other views of professional inquiry designed to help 
teachers not only develop new knowledge and skills but interrogate their views and 
question and critique the political nature of schooling as it relates to students 
learning mathematics (e.g., Gutierrez, 2007; Horn, 2005; King, 2002). This more 
critical stance of inquiry is an attempt to support teachers to consider how schools 
and learning opportunities affect access and equity in students' academic identities 
and trajectories in successfully learning mathematics and pursuing mathematics as 
they continue through schools. 

Viewing schools as sites for teacher learning rather than as places where 
teachers simply work is well supported by sociocultural theories of learning. 
Drawing from the notion that knowledge and meaning are constructed through 
practice, Franke, Carpenter, Fennema, Ansell, and Behrend (1998, p. 68) argue for 
supporting learning that is self-sustaining and generative; they claim that teachers' 
supported efforts to engage in classroom practices guided by student learning can 
serve as a basis for their own continued growth and problem solving of classroom 

It is in developing an understanding of their practices in relation to their 
students' learning that teachers develop the understanding necessary to 
generate new ideas. If a teacher struggles to understand why the students are 
successful, how they are solving problems, how their thinking develops, and 
how instruction might help students to build on their current conceptions, 
connections are made, understanding develops, and the potential for more 



connections becomes possible. Thus, there exists a basis for the teacher to 
learn and continue to grow. 

Through Franke and colleagues' work, we have learned how teachers can use 
their own practices to make sense of student learning. However, we need to 
understand further how teachers' classroom practices and broader school-based 
professional communities can provide the basis for teachers to continue to develop 
their practice. In their work, Franke and colleagues argued that teachers who are 
generative develop detailed understanding of their own students' thinking, organize 
that knowledge and view it as their own to create, adapt, and change. These 
teachers learn from interacting with their students; they are focused in the ways 
they listen to, interpret, and make use of their students' thinking. In this chapter on 
school development, 1 consider how teachers' participation in multiple and 
potentially overlapping communities of practice shapes and re-shapes their 
identities and constitutive skills and knowledge as teachers of mathematics. 


The main mechanism for supporting teachers' generative learning for ambitious 
teaching has been through building and sustaining professional communities of 
teachers. Theorizing teacher learning through participation in professional 
community draws on sociocultural theories of learning which take participation as 
a key construct (see also Jaworski, this volume). Lave and Wenger (1991) define a 
community of practice as "a set of relations among persons, activity and world, 
over time and in relation with other tangential and overlapping communities of 
practice" (p. 98). A community of practice can be defined through mutual 
engagement in a joint enterprise which develops shared repertoires of practice 
(Wenger, 1998). Lave (1996) describes learning as "changing participation in 
changing 'communities of practice'" (p. 150). Learning is not a process of 
acquiring or transmitting knowledge. Rather learning is apparent in the way 
participation transforms within a community of practice (Rogoff, 1997). The shifts 
in participation do not merely mark a change in a participant's activity or 
behaviour; a shift in participation also involves a transformation of roles and the 
crafting of a new identity, one that is linked to but not completely determined by 
new knowledge and skills (Lave, 1996; Lave & Wenger, 1991; Rogoff, 1994, 
1997; Wenger, 1998). Lave (1996, p. 157) states, "crafting identities is a social 
process, and becoming more knowledgeably skilled is an aspect of participation in 
a social practice. By this reasoning, who you are becoming shapes crucially and 
fundamentally what you 'know'." Knowledge and the development of skill are 
clearly important in understanding learning. Developing skill and knowledge is in 
service of changing participation in a particular community. 

Lave and Wenger (1991) describe transforming participation in terms of 
movement from legitimate peripheral to full participation. As a legitimate 
participant, one is connected and belongs to the community of practice in question, 
but as a peripheral participant, one engages less fully in the community. The 



peripheral participant has access and can move towards full participation, thus 
developing an identity of full participation. Full participation entails "developing 
an identity as a member of a community and becoming knowledgeably skillful" 
(Lave, 1991, p. 65). Analysis of learning focuses on the structuring of the 
community's work practices and learning resources; learning is detectable in 
members' participation in the work of the community. 

What Does Professional Community Mean for Mathematics Teaching and 

As mathematics education researchers have drawn on the community of practice 
theory, they have identified key features of professional community, the strength of 
which inspires instructional innovation and commitment to students. Several 
reviews exist which compare these features and common to all these communities 
is a shared sense of purpose and collective and coordinated collaborative activity 
with a commitment to students (e.g., Dean & McClain, 2006; Sowder, 2007). Dean 
and McClain (2006, pp. 13-14) define these as: 

A shared purpose or enterprise such as: ensuring that students come to 
understand central mathematical ideas while simultaneously performing more 
than adequately on high stakes assessments of mathematics achievement. 

A shared repertoire of ways of reasoning with tools and artefacts that is 
specific to the community and the shared purpose including normative ways 
of reasoning with instructional materials and other resources when planning 
for instruction or using tasks and other resources to make students' 
mathematical reasoning visible. 

Norms of mutual engagement encompassing both general norms of 
participation as well as norms specific to mathematics teaching such as the 
standards to which the members of the community hold each other 
accountable when they justify pedagogical decisions and judgments. 

Gutierrez (1996) calls high school mathematics departments with strong 
professional communities as "organized for advancement'' because their collective 
activity makes a difference in advancing student learning and achievement. In 
addition, several researchers emphasize teachers' ability to wield influence and 
control over important decisions that affect a school's activities, policies, and 
curriculum (Erickson, Brandes, Mitchell, & Mitchell, 2005; King, 2002; Little, 
1999; Secada & Adajian, 1997). 

Cultivating Professional Inquiry at the School Level 

This chapter underscores the idea that school development necessarily involves 
learning. Boreham and Morgan (2004) argue that organizations learn because 
members of the community are able to coordinate their perspectives and actions 



towards achievement of common goals. They found that organizations learn by 
developing relational practices, "the kind of practice[s] by which people connect 
with other people in their world, and which direct them to interact in particular 
ways" (p. 3 1 5). These relational practices include: 

- opening space for creation of shared meaning 

- reconstituting power relations 

- providing cultural tools to mediate learning. 

Those relational practices can be connected to developing social and intellectual 
resources (opening space for creation of shared meaning and reconstituting power 
relations) and material resources (providing cultural tools to mediate learning). In 
what follows, I draw on the professional community and inquiry literature to 
identify the following dimensions that make possible these relational practices. 

- Activating school leaders 

- Navigating fault lines, dealing with micropolitical issues among teachers 

- Developing and sustaining a focus 

- Engaging parents as intellectual and social resources 

Activating school leaders. When the school is the centre of change, theories of 
action for supporting professional learning communities (PLCs) among 
mathematics teachers necessitate contention with school cultures and institutional 
realities. Gamoran and colleagues (2003) contend that PLCs need access to 
resources - material, human, and social - if they are to remain viable. Here the role 
of school leaders in facilitating the availability and use of such resources can be 
critical. School leaders refer not only to school principals or heads but also to 
curriculum leaders, mathematics coaches, teacher mentors, or faculty coordinators. 
There is variance documented in the literature as to how much direct involvement 
in a professional community principals have versus other mathematics leaders 
(Burch & Spillane, 2003; King, 2002; Krainer & Peter-Koop, 2003; Wolf, Borko, 
Elliott, & Mclver, 2000). In some cases, principals participate in teacher meetings 
in order to learn alongside teachers and spend time in classrooms. In others, the 
principal is supportive by allocating school level resources in providing space and 
time and by making goals of teacher interactions congruent with school and district 
level goals (Coburn & Russell, in press). Whether and how leadership strategies 
interact with supporting professional development in a PLC at the school level in 
mathematics, is a burgeoning area of research. Findings underscore the situated 
way in which leadership strategies interact with local conditions. 

Halverson (2007) documented the way school leaders make use of certain 
structures to enable the building of a professional community. Key to Halverson's 
(2007) analysis is that a professional community is a "form of organizational trust" 
(p. 94) resulting from the kinds of interactions teachers have to consider in using 
alternative instructional strategies to improve student learning. His analysis of the 
role of school leaders in supporting a professional community focuses in part on 
leaders' use of artefacts, such as role positions, daily schedules, meetings and 
meeting agendas. Moreover, he found that school leaders sequence the use of these 



artefacts in different ways to initiate interactions, facilitate development of mutual 
obligations, and provide feedback about how those obligations are being met at a 
systemic level. One of the key material resources in supporting teachers' ability to 
work together is time and space and to do so in the rushed pace of the normal work 
week. Halverson found that leaders strategically made use of local contingencies in 
order to carve out space and time for teachers to initiate and legitimate time to talk 
about instruction - in one case through Breakfast Club meetings and in two others 
through grant-writing projects to respond to a new accountability mandate. 
Examples abound in the literature of other means that school leaders employ to 
initiate conversations, including such things as analysis of student work (e.g., 
Kazemi & Franke, 2004), implementing common units of lessons (e.g., Borasi, 
Fonzi, Smith, & Rose, 1999), study groups (e.g., Arbaugh, 2003), and regular 
department meetings (e.g., Horn, 2005). Coburn and Russell's (in press) use of 
social network analysis also supports the importance of school leaders in allocating 
human resources such as coaching expertise in collective professional interaction. 
They found that the allocation of coaches affected depth of interactions in PLCs. 

Navigating fault lines. Establishing professional communities is not just a matter 
of decreeing that one exists. There is nothing neither positive nor harmonious 
inherent in the term community, and if we consider the countless times researchers 
have characterized the work of teaching and school reform as complex, we should 
expect that forming inquiry communities within schools should not be a trivial 
matter. Research on PLC and school-based reform has highlighted how managing 
tensions and conflicts are critical for the viability of the PLC (Rousseau, 2004). 
Grossman, Wineburg, and Wool worth (2001) identified the navigation of fault 
lines as one of the central concerns of building a supportive teacher community. 
These fault lines are not necessarily interpersonal conflicts among teachers but may 
be tensions regarding disciplinary goals and views of teaching and learning 
mathematics that exist in the school or department (e.g., Rousseau, 2004). Because 
participating in a PLC is about changing school cultures and breaking norms of 
privacy while at the same time building norms of critical colleagueship, 
researchers have also attended to the ways such participation affects 
transformations in teacher identity (e.g., Drake, 2006; Kelchtermans, 2005; Battey 
& Franke, in press). Researchers argue that teachers' individual reactions to a 
PLC's demands are mediated by social and cultural contexts as well as teachers' 
working dynamic identities. "Teachers' identities carry personal histories, emotion, 
values, and knowledge and they shape how teachers participate in professional 
development and their classrooms" (Battey & Franke, in press, p. 27). 

Research has discussed the paradoxes and conflicts that are bound up in 
cultivating PLCs. Many researchers have noted that in order for teacher inquiry to 
have a school-wide effect, it must move beyond individual teachers. But mandating 
participation in teacher inquiry at the school-level can also backfire. Here 
leadership must be strategic in inviting a critical mass of participation that can have 
a pronounced affect on school culture (Berger, Boles, & Troen, 2005; Krainer, 



2001). The role of key school leaders (whether principals, school facilitators, or 
coaches), again becomes critical in creating a press for teacher inquiry and in 
developing collective ownership (Berger et al., 2005; Nickerson & Moriarty, 

Developing and sustaining a focus. If schools are successful in carving out time 
and space for teachers to meet together, in managing conflicts and tensions, the 
next dimension I wish to highlight is how researchers describe the importance of 
developing and sustaining a focus on students. Because time and space are precious 
resources in schools, collegial interactions need to be focused in order to be 
productive. Some studies indicate that material resources such as curriculum units 
or lessons, student work, videotaped lessons can be leveraged successfully in order 
to achieve this focus (e.g., Borasi et al., 1999; Kazemi & Franke, 2004; Seago, this 
volume; Sherin, 2004). Lin's (2002) study is illustrative. The school-based 
professional inquiry she described focused on grade-level collaborations. First- 
grade teachers met to develop, observe, and reflect on lessons from the Taiwanese 
mandated learner-oriented reform curriculum. To provide sufficient focus for their 
collective inquiry, the first-grade teachers selected a common lesson to plan and 
observe in each other's classrooms. They used these common lessons and 
observations to write classroom cases that in effect focused and deepened their 
conversations, beyond what observation alone would have accomplished. Three 
types of cases emerged from this collaborative inquiry: (1) analysis of students' 
varied solutions and strategies; (2) students' interpretations of other students' 
thinking; and (3) comparisons of two different instructional approaches to teaching 
the same topic. Cases were developed in several phases; each successive phase 
refined and elaborated the teaching context and the questions for discussions. The 
creation of teaching cases situated in teachers' own classrooms when all teachers 
were able to teach the same lesson enabled the teachers to very carefully analyse 
the effect of task design and sequencing on students' mathematical thinking; 
students' ability to use symbolic and pictorial representations to solve problems; 
and for teachers to compare and think deeply about how their instructional choices 
(e.g., "Is binding straws important for students' ability to count by tens?") 
impacted student performance. This was dependent, of course, on teachers' 
willingness to examine their own practice and to raise questions for one another. 
This kind of inquiry is dependent on cultivating norms of inquiry, navigating fault 
lines, and developing resources for time to meet. 

Engaging parents as intellectual and social resources. The role of parents in 
school development has received much less attention than the work of teachers, 
administrators, and leaders. This makes sense given the amount of attention of 
coordinating professionals who work at the school itself. Nonetheless, recent work 
on parents has illuminated a number of issues related to parents' engagement, 
raising possible considerations for parents' role in supporting school development. 
Improving mathematics instruction involves a dramatic transformation from 
viewing mathematics as a fixed body of procedures to memorize and apply with 



fidelity to a discipline that is fundamentally about complex problem solving, 
justification, and argumentation. Studies investigating parents' views have revealed 
that parents can feel disempowered in relation to reform-oriented mathematics 
(Martin, 2006; Remillard & Jackson, 2006). At the same time, interventions that 
have aimed to work with families around mathematics as a complex problem 
solving discipline have reported significant increases in families' feelings of 
empowerment (Civil & Bernier, 2006). Reports of building school capacity may 
include events such as parent nights designed to mitigate these anxieties by 
beginning dialogue with families about goals of ambitious mathematics teaching 
and learning (e.g., Kramer & Keller, 2008). We have much more to learn about 
how engagement with families can support school development efforts, in what 
kinds of social and political contexts, and to what ends. Might leadership, for 
example, leverage family views, practices and questions to catapult teacher inquiry 
(e.g., Anderson, 2006) or could family practices undercut professional community? 

Studying the Practices of School-Based Professional Communities 

Cultivating a viable professional community naturally begs questions about how 
members of the school community actually interact with one another as they 
discuss teaching practice. What might it look like for teachers and leaders to 
engage in critical colleagueship (Lord, 1994) oriented to improving mathematics 
teaching and learning? I find the work of Little and Horn (2007) to be particularly 
instructive in thinking about the practices of a professional community. They are 
developing a conceptual scheme to examine what actually happens in collective 
professional learning communities. How do participants actually work together? 
What do they say and do? How do they interact with one another around artefacts 
of practice and how do they talk about classroom instruction? Little (2002) offers 
two central questions for examining interactions within teacher communities: 

1 . What faces of practice are made visible through talk and with what 
degree of transparency? 

2. How does interaction open up or close down opportunities to learn? 

The face of practice refers to "those parts of practice that come to be described, 
demonstrated, or otherwise rendered in public exchanges among teachers" (Little, 
2002, p. 934), which may include artefacts such as student work. Transparency of 
practice conveys "how fully, completely, and specifically various parts of practice 
are made visible or transparent in the interaction" (Little, 2002, p. 934). These two 
questions seem central to us in order to understand what views of practice are made 
available to teachers through their collective inquiry. As I describe below, I 
contend that the understanding of the inner workings of school-based professional 
development communities should be then related to teachers' participation in and 
out of the group setting. This relational view can help us understand who teachers 
are becoming through this process (Battey & Franke, in press; Enyedy, Goldberg, 



& Welsh, 2006), and how they are enacting their developing identities, skills, and 
knowledge with their students in the classroom. 

Little and Horn (2007; see also Horn, 2005) have documented in detail how the 
rendering of classroom interactions in professional conversations shape 
opportunities for teacher learning. By contrasting informal conversations in 
mathematics department meetings in two different high schools, they compare how 
teachers use replays and rehearsals to reason publicly about their instructional 
practice and to consider alternative interpretations and re-formulations of 
pedagogical dilemmas and problems in ways that propel their teaching forward. 
During replays, teachers recount "blow-by-blow accounts of classroom events, 
often acting out both the teacher and students' roles" (Horn, 2005, p. 225). 
Through rehearsals, teachers act out classroom interactions that might occur in the 
future, anticipating what they might say and how students might respond. In 
careful analysis of teacher-to-teacher talk, Horn documents how teachers move in 
and out of these modes as they reconsider their pedagogical choices and ready 
themselves for continued experimentation. However, the presence of replays and 
rehearsals during collegial interactions is not sufficient for such experimentation. 
How such replays and rehearsals function in framing pedagogical issues, questions, 
dilemmas and frustrations is what matters. 

Prior work by Kazemi and Franke (2004; see also Franke & Kazemi, 2001; 
Franke, Kazemi, Shih, Biagetti, & Battey, 2005) documented detailed images of 
the evolution of collective inquiry among elementary teachers analysing their 
students' mathematical work in monthly school-based meetings in which each 
teacher had posed a similar mathematical problem (focused on number and 
operations) to his or her class. The analysis of the teacher talk revealed that the 
workgroup conversations evolved as teachers learned to talk about their work. 
Salient developments included the following: 

- Teachers first had to leam to attend to the details of their students' thinking. 
Even though meetings were structured from the very beginning to detail 
students' strategies, teachers did not come to the first meeting prepared to do so. 
Instead, many teachers assumed that the pieces of paper themselves would tell 
the story. It was further evident in the way they posed the problems that many 
assumed that conversations with students about their solutions were not 
necessary. Maintaining the structure of the workgroup promoted an emphasis on 
documenting the details of student thinking. Kazemi and Franke (2004) 
intentionally facilitated the discussions so that teachers would return to the idea 
of noticing how students were learning to break apart and put together numbers 
using their knowledge of the base ten structure of the number system. In order 
for teachers to interpret their students' reasoning, they began to use the student 
work as a trace rather than a complete record of their students' reasoning. 
Without such a mathematical focus, meetings may not encourage teachers to 
follow a particular course of experimentation in their classrooms. 

- Teachers' close consideration of student reasoning opened up opportunities to 
deliberate mathematical and pedagogical questions. Examining student work, 
as structured in this professional development, allowed teachers to surface their 



confusions and uncertainties, not just about student reasoning but also about 
mathematics and classroom practice. The discussions opened up opportunities 
for teachers to notice the mathematical ideas students were using. This led to the 
group's engagement with sophisticated computational strategies they were 
noticing in their classrooms. The use of student work provided an entry point for 
teachers to explore mathematical ideas and have opportunities to make sense of 
efficient student-generated algorithms. Pedagogical issues related to helping 
children develop more sophisticated strategies also surfaced once the group saw 
students in teachers' own classrooms using such strategies. 
- Diversity in teachers' experimentation served as a resource for learning. 
Teachers differed in how they reported on their engagement with students in 
their classrooms. Not all the teachers experimented with these ideas in the same 
way. Because the group had multiple ways of relating the professional 
development discussions to their classroom practices, the experiences of some 
teachers generated new conjectures about what to try in the classroom. The 
frustrations teachers shared in the group also underscored the challenges they 
faced in helping children articulate and build their ideas. This diversity became 
a resource for teachers to compare and question each other's practices. 
Understanding how teachers' interact with one another in PLCs and how those 
interactions evolve and develop learning opportunities for teachers is vital for 
research and development work in fostering collective inquiry at the school level. 
In the next subchapter, I argue for relating this evolving research base to the well- 
known research on teachers' classroom instruction. 


Earlier in this chapter, I proposed the view that school development deals with the 
development of organizational supports for ambitious mathematics teaching and 
learning. Much of the literature emphasizes mechanisms for such improvement to 
be highly dependent on how schools employ resources and how they are organized 
to support teachers' collective learning together outside of the classroom. What I 
would like to offer in the remainder of this chapter is a proposal for attending to 
one aspect of the relationship between joint inquiry at the school level and 
classroom instruction. My discussion is not meant to be comprehensive; it 
identifies one shortcoming of how we have typically thought about the impact of 
teachers' and leaders' joint inquiry on mathematics instruction and student 
learning. Some of our current research base is limited in scope to a unidirectional 
view of the impact of cultivating professional inquiry at the school level to 
teachers' classroom practice. I propose that what needs to be addressed and 
developed are ways to characterize and document the multi-directional influence 
between participation in joint inquiry and the individuality of classroom practice. 
In this last subchapter, I outline the rationale for this approach and describe some 
of the necessary theoretical and empirical work that lies ahead. 



Teachers are simultaneously involved in multiple activity settings, including 
their own classroom, school, and district. When they are involved in sustained 
inquiry with colleagues and leaders at the school, this constitutes another important 
mediating context. I use the construct of activity settings to focus on the boundaries 
and relationships between the classroom, school and district (see also Cobb & 
Smith, this volume). Activity settings overlap; that is, they do not exist as insular 
social contexts but rather as sets of relationships that coexist with others 
(EngestrSm, 2001). Activity settings can have temporal, conceptual, and physical 
boundaries (Grossman, Smagorinsky, & Valencia, 1999). It is this dynamism 
across activity settings and how that shapes teacher learning that I am concerned 
with in this chapter. To develop these ideas further and specify a way of talking 
about learning, I draw on Cook and Brown's (1999) distinction between knowledge 
and knowing. Knowledge, in their view, is something that we "possess". We 
"deploy" this knowledge in our actions. In their words, "Knowing refers to the 
epistemic work that is done as part of action or practice, like that done in actual 
riding of a bicycle or the actual making of a medical diagnosis" (p. 387). 
Knowledge, then, can be seen a tool of action because individuals or groups can 
use knowledge (whether tacit or explicit) to discipline their interactions with the 
world. This distinction seems both relevant and important in thinking about teacher 
learning. Much has been written about the kinds of specialized knowledge that 
teachers need, among them, knowledge of the discipline, their students, and 
instructional strategies (Ball & Bass, 2000; Shulman, 1986). Professional inquiry 
clearly needs to develop teachers' knowledge, and we have been rightfully 
concerned with figuring out what kinds of knowledge teachers gain through 

Cook and Brown (1999) would agree that knowledge is essential for practice but 
it is not sufficient for explaining what it takes to be good at what you do: "An 
accomplished engineer may possess a great deal of sophisticated knowledge; but 
there are plenty of people who possess such knowledge yet do not excel as 
engineers" (p. 387). In addition to all the kinds of knowledge that teachers need, 
they also have to be able to teach. For me, this means that we have to attend to the 
interplay between knowledge and knowing in the professional community and in 
teachers' classroom context. In addition, we need to attend to the interplay between 
teachers' development of knowledge and ways of knowing in the professional 
development and classroom contexts over time. We need to link the knowledge and 
ways of knowing that teachers develop together with what happens as teachers try 
to use the knowledge and ways of knowing they gain in joint inquiry in the context 
of their classroom teaching. Teachers may develop similar ways of examining and 
talking about students' mathematical thinking through inquiry with colleagues in 
their school but we clearly need to concern ourselves with how they are drawing on 
that knowledge when they interact with students, or in Cook and Brown's terms, 
how knowledge is deployed in the service of disciplining action (knowing). 
Moreover, I argue that researchers should examine what teachers are learning 
during and after conversations with colleagues, looking at the coevolution of 
participation between classroom practice and professional inquiry. I claim that this 



coevolution between the contexts of professional collaboration and classrooms 
should itself be a key unit of analysis as we try to explicate the mechanisms by 
which teachers learn in and through professional inquiry. By seeing how teachers' 
participation across these contexts co-evolves, we will have better views of the 
relationship between joint professional inquiry, learning and instruction, and school 

The Implications of a Multidirectional Analysis for Studying and Designing 
Collective Inquiry 

A multidirectional analysis of professional learning across contexts where teachers 
and leaders work together and the classroom leads us to the following implications 
for the study and design of these efforts. We should: (1) understand and elicit the 
diversity of teachers' experimentation and the effect of depictions of that work in 
joint inquiry and (2) examine the situated nature of primary artefacts. 

Understanding the diversity of teachers ' experimentation. In order to understand 
the relationship between the contexts of joint inquiry and the classroom, and how 
teachers' and leaders' participation across these settings co-evolves, we must 
understand individual teachers' classroom experimentation, and how this 
influences their collaboration with colleagues. How do teachers deploy their 
knowledge in the classroom? What ways of knowing do they demonstrate in their 
instructional practice? What do teachers bring to the collective as a result of their 
experimentation? In addition to documenting the diversity of individual teachers' 
classroom experimentation, we also need to document and study what actually 
happens when teachers and leaders come together and their collective learning 
trajectories as they participate in this context. It is essential that we document the 
diversity of teachers' classroom experimentation and study the nature of how this 
experimentation relates to their experiences over time with their colleagues and to 
their developing identities - what kinds of teachers do they want to become? What 
ways of knowing are developed over time? How and what knowledge do teachers 
develop of subject matter, students' thinking, and practice as they engage in 
collective analysis around common objects of inquiry? 

While the argument here is about research on teachers' joint learning, there are 
also implications for the design of collective inquiry (see also Jaworski, this 
volume; Seago, this volume). I argue not only that teachers' experimentation 
should be studied but that leaders of teachers' joint inquiry should incorporate 
depictions of teachers' classroom experimentation in collaborative engagement. 
Depictions of practice are images or stories that seek to capture the events in the 
classroom as they played out, earlier referenced as replays and rehearsals. They are 
created intentionally to support the analysis of teaching. Written cases and video- 
cases are perhaps the most visible example of depictions available in the literature. 
But there are other examples: replays and rehearsals (Horn, 2005) are depictions 



that are created through teachers' talk; teachers' journals can also serve as a 

If professional educators sought openings to elicit teachers' experimentation in a 
principled way, collective inquiry could serve as a place to pursue questions and 
dilemmas teachers encounter as they engage in transforming their practice. While it 
is easy to advocate that we incorporate depictions of practice and discuss teachers' 
classroom experimentation in the context of collective inquiry more extensively, I 
recognize that sharing episodes from the classroom can easily and unproductively 
spiral into a show-and-tell. Leaders of collective inquiry will need to become more 
knowledgeable and skilled about how to use teachers' classroom experiences. For 
example, how can the dilemmas teachers face about modifying tasks, managing 
pacing, and orchestrating classroom discourse be usefully depicted and used as a 
springboard for discussion? How can leaders utilize one teacher's experiences to 
support another to develop more focused and reflective attempts to experiment in 
the classroom? Many researchers have written extensively about the intentional use 
of records of practice (e.g., Sherin, 2004; Lampert & Ball, 1998; Little, 2004), 
arguing that we must attend not only to the careful selection of representations but 
also how they are negotiated in practice. 

Examining the situated nature of primary artefacts. Primary artefacts are objects 
that originate (or are produced for use) in instructional practice. In the case of 
teaching, primary artefacts include copies of student work, lesson plans, 
mathematical tasks, and curriculum materials. They can travel across boundaries, 
into contexts where teachers and leaders collaborate, but they are not created solely 
for the purpose of collectively analysing teaching. Primary artefacts allow 
particular components of teaching to be extracted from the context of instructional 
practice, lessening the complexity by narrowing teachers' focus. 

Primary artefacts are produced and used in practice, and so ways of knowing 
include the use and production of primary artefacts. If we are concerned with 
teachers developing new ways of knowing in their classroom practice, then we 
should attend to the relationship between ways of knowing in professional 
development and in the classroom. And if we are going to use primary artefacts as 
a tool in professional development, we must attend to how they are situated in 
particular activities, and how this affects their meaning. For example, student work 
is a primary artefact commonly used in collective work of teachers. The way 
student work is situated in collective inquiry may look very different from its use 
in the classroom. Teachers and leaders may sit together to analyse a collection of 
pre-selected student work to illustrate the range of solution strategies students used 
in their classroom. They may spend extended time debating what students 
understand, generating questions they might ask to better understand the students' 
thinking, or considering which strategies they would choose to highlight in a whole 
class discussion. In contrast, in their classrooms, teachers may only have a few 
minutes to survey students' written work in order to make assessments and 
instructional decisions. The teacher typically engages in this work alone, in the 
midst of a lesson, while students are working on the task. While collective inquiry 



may certainly help teachers gain knowledge they can deploy in this classroom 
situation, it may not help them develop the ways of knowing they need to monitor 
students in the moment and to interact with them in ways that assess and advance 
students' mathematical thinking. Researchers and leaders must attend to the 
meaning teacher's make of primary artefacts across contexts as these artefacts are 
situated in different activities. We need to better understand how the ways of 
knowing involved in these activities differ, and how they influence one another. 


Writing about school development is necessarily a synthetic enterprise. In this 
chapter, I conceptualized school development as a school's efforts to support 
ambitious teaching and learning. Specifically, I took a learning perspective on 
school development. This perspective highlights how schools support professional 
learning. Collinson (in press, p. 7) states: 

Vibrant, innovative organizations work at developing their organizational 
capacity by establishing an environment in which members, and thereby the 
organization, can continuously learn and improve. Developing members, 
along with careful recruiting and hiring of fresh talent, ensures innovation 
and renewal. 

The way we understand and study professional learning in schools has 
significant implications for the way we structure and support teachers' collective 
learning opportunities, the goals we create for inquiry, and as researchers and 
educators, the ways we collaborate with schools to improve mathematics teaching 
and learning. To understand teacher collective learning within the context of 
school-based professional development, I have argued that we need to develop 
conceptual frameworks to take into account both the dynamics of individual 
teacher learning and vulnerabilities to developing their instructional practices as 
well as the resources and settings that support learning. 

One noteworthy aspect of writing this chapter on school development was the 
opportunity to review research that draws on both classroom level research and 
organizational and policy implementation research. New collaborations forged 
between classroom researchers and policy or organizational researchers (e.g., Cobb 
& Smith, 2007; Cobb & Smith, this volume; Gamoran et al., 2003; Little & Horn, 
2007) can enrich our view of designing for a tighter coupling between teacher 
learning and whole school development that would ultimately benefit professional 
culture within the school and students' mathematical learning. That said - here are 
a number of issues that remain to be incorporated into our studies of and designs 
for school development. 

- We need to explicate significant differences between working with schools at 
the primary and secondary level and how those impact the ways schools support 
ambitious teaching and learning. There seem to be different challenges with 
respect to teachers' skills and identities as mathematics teachers, tracking or 
grouping practices, testing practices and their consequences, curriculum 



organization, and relationships among teachers, school leaders, and parents 
(e.g., Lee, Smith, & Croninger, 1997; Spade, Columba, & Vanfossen, 1997). 

- As was evident in this chapter, the research literature on school development 
attends predominantly to the role of teachers and school leaders. How parents 
and families figure into school development and supporting ambitious teaching 
remains underdeveloped and undertheorized. 

- Our understanding of school development can be strengthened through further 
study of how prospective and novice teachers are involved in collective inquiry 
as a way to recruit and continue to develop practices of the school community 
(see Leikin, this volume). 

Our field's ability to address these issues and others over the next decade will 
advance our understanding of school development and inform the next generation 
of interventions aimed at supporting ambitious teaching. 


I would like to thank Allison Hintz for her invaluable assistance identifying 
research used in the literature review. Work with Megan Franke, Amanda 
Hubbard, and Magdalene Lam pert has been instrumental in developing some of the 
theoretical ideas. I am grateful for the helpful comments of Paul Cobb, Gilah 
Leder, Heinz Steinbring, and Terry Wood on previous versions of this chapter. I 
am especially indebted to Konrad Kramer for his help in formulating the focus of 
this chapter. 


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Elham Kazemi 
College of Education 
University of Washington 






This chapter focuses on research that can inform the improvement of mathematics 
teaching and learning at scale. In educational contexts, improvement at scale 
refers to the process of taking an instructional innovation that has proved effective 
in supporting students ' learning in a small number of classrooms and reproducing 
that success in a large number of classrooms. We first argue that such research 
should view mathematics teachers' instructional practices as situated in the 
institutional settings of the schools and broader administrative jurisdictions in 
which they work We then discuss a series of hypotheses about structures that 
might support teachers ' ongoing improvement of their classroom practices. These 
support structures range from teacher networks whose activities focus on 
instructional issues to relations of assistance and accountability between teachers, 
school leaders, and leaders of broader administrative jurisdictions. In describing 
support structures, we also attend to equity in students' access to high quality 
instruction by considering both the tracking or grouping of students in terms of 
current achievement and the category systems that teachers and administrators use 
for classifying students. In the latter part of the chapter, we outline an analytic 
approach for documenting the institutional setting of mathematics teaching that 
can feed back to inform instructional improvement efforts at scale. 


In educational contexts, improvement at scale refers to the process of taking an 
instructional innovation that has proved effective in supporting students' learning 
in a small number of classrooms and reproducing that success in a large number of 
classrooms. In countries with centralized educational systems, it might be feasible 
to propose taking an instructional innovation to scale at the national level. 
However, proposals for instructional improvement at the national level are usually 
impractical in countries with decentralized education systems because the 
infrastructure that would be needed to support coordinated improvement at the 
national level does not exist. The case of instructional improvement at scale that 
we consider in this chapter is located in a country with a decentralized education 

1 The analysis reported in this chapter was supported by the National Science Foundation under grant 
No. ESI 0554535. The opinions expressed do not necessarily reflect the views of the Foundation. The 
hypotheses that we discuss in this chapter were developed in collaboration with Sarah Green, Erin 
Henrick, Chuck Munter, John Murphy, Jana Visnovska, and Qing Zhao. We are grateful to Kara 
Jackson for her constructive comments on a previous draft of this chapter. 

K. Krainerand T. Wood (eds.J, Participants in Mathematics Teacher Education, 23 1-253. 
© 2008 Sense Publishers. All rights reserved. 


system, the US, in which there is a long history of local control of schooling. Each 
US state is divided into a number of independent school districts. In rural areas, 
districts might serve less than 1 ,000 students whereas a number of urban districts 
serve more than 100,000 students. In the context of the US educational system, 
when we speak of scale we have in mind the improvement of mathematics teaching 
and learning in urban districts as they are the largest jurisdictions in which it is 
feasible to design for improvement in the quality of instruction (Supovitz, 2006). 
In this chapter, we speak of instructional improvement at the level of the school 
and the district with the understanding that the appropriate organizational unit or 
administrative jurisdiction beyond the school needs to be adjusted depending on 
the structure of the educational system in a particular country. 

The central problem that we address in this chapter is how mathematics 
education research can generate knowledge that contributes to the ongoing 
improvement of mathematics teaching and learning at scale. The daunting nature 
of "the problem of scale" is indicated by the well-documented finding that prior 
large-scale improvement efforts in mathematics and other subject matter areas 
have rarely produced lasting changes in either teachers' instructional practices or 
the organization of schools (Elmore, 2004; Gamoran, Anderson, Quiroz, Secada, 
Williams, & Ashman, 2003). Schools frequently experience external pressure to 
change, a condition that Hesse (1999) has termed policy churn. However, in most 
countries, classroom teaching and learning processes have proven to be remarkably 
stable amidst the flux. Cuban (1988), a historian of education, likened the situation 
to that of an ocean tossed by a storm in which all is calm on the sea floor even as 
the tempest whips up waves at the surface. 

Researchers who work closely with teachers to support and understand their 
learning will probably not be surprised by Elmore's (1996) succinct synopsis of the 
results of educational policy research on large-scale reform: the closer that an 
instructional innovation gets to the core of what takes place between teachers and 
students in classrooms, the less likely it is that it will implemented and sustained 
on a large scale. This policy research emphasizes that although research-based 
curricula and high-quality teacher professional development are necessary, they are 
not sufficient to support the improvement of mathematics instruction at scale. 
Instructional improvement at scale also has to be framed as a problem of 
organizational learning for schools and larger administrative jurisdictions such as 
districts (Blumenfeld, Fishman, Krajcik, Marx, & Soloway, 2000; Coburn, 2003; 
McLaughlin & Mitra, 2004; Stein, 2004; Tyack & Tobin, 1995). This in turn 
implies that in addition to developing new approaches for supporting students' and 
teachers' learning, reformers also need to view themselves as institution-changing 
agents who seek to influence the institutional settings in which teachers develop 
and refine their instructional practices (Elmore, 1996; Stein, 2004). We capitalize 
on this insight in our chapter by emphasizing the importance of coming to view 
mathematics teachers' instructional practices as situated within the institutional 
setting of the school and larger jurisdictions such as districts. This perspective 
implies that supporting teachers' improvement of their instructional practices 
requires changing these settings in fundamental ways. 



In the US context, the institutional setting of mathematics teaching, as we 
conceptualize it, encompasses district and school policies for instruction in 
mathematics. It therefore includes both the adoption of curriculum materials and 
guidelines for the use of those materials (e.g., pacing guides that specify a timeline 
for completing instructional units) (Ferrini-Mundy & Floden, 2007; Remillard, 
2005; Stein & Kim, 2006). The institutional setting also includes the people to 
whom teachers are accountable and what they are held accountable for (e.g., 
expectations for the structure of lessons, the nature of students' engagement, and 
assessed progress of students' learning) (Cobb & McClain, 2006; Elmore, 2004). 
In addition, the institutional setting includes social supports that give teachers 
access to new tools and forms of knowledge (e.g., opportunities to participate in 
formal professional development activities and in informal professional networks, 
assistance from a school-based mathematics coach or a principal who is an 
effective instructional leader) (Bryk & Schneider, 2002; Coburn, 2001; Cohen & 
Hill, 2000; Horn, 2005; Nelson & Sassi, 2005), as well as incentives for teachers to 
take advantage of these social supports. 

The findings of a substantial and growing number of studies document that 
teachers' instructional practices are partially constituted by the materials and 
resources that they use in their classroom practice, the institutional constraints that 
they attempt to satisfy, and the formal and informal sources of assistance on which 
they draw (Cobb, McClain, Lamberg, & Dean, 2003; Coburn, 2005; Spillane, 
2005; Stein & Spillane, 2005). The findings of these studies call into question an 
implicit assumption that underpins many reform efforts, that teachers are 
autonomous agents in their classrooms who are unaffected by what takes place 
outside the classroom door (e.g., Krainer, 2005). In making this assumption, 
reformers are, in a very real sense, flying blind with little if any knowledge of how 
to adjust to the settings in which they are working as they collaborate with teachers 
to support their learning. In contrast, the empirical finding that teachers' 
instructional practices are partially constituted by the settings in which they work 
orients us to anticipate and plan for the school support structures that need to be 
developed to support and sustain teachers' ongoing learning. 


One of the primary goals of our current research, which is still in its early stages, is 
to generate knowledge that can inform the ongoing improvement of mathematics 
teaching and learning at scale. To this end, we are collaborating with four large, 
urban districts that have formulated and are implementing comprehensive 
initiatives for improving the teaching and learning of middle-school mathematics. 
We will follow 30 middle-school mathematics teachers and approximately 17 
instructional leaders in each of the four districts for four years to understand how 
the districts' instructional improvement initiatives are playing out in practice. In 
doing so, we will conduct one round of data collection and analysis in each district 
each year for four years to document: 1) the institutional setting of mathematics 
teaching, including formal and informal leaders' instructional leadership practices, 



2) the quality of the professional development activities in which the teachers 
participate, 3) the teachers' instructional practices and mathematical knowledge for 
teaching, and 4) student mathematics achievement. The resulting longitudinal data 
on 120 teachers and approximately 68 school and district leaders in 24 schools in 
four districts will enable us to test a series of hypotheses that we have developed 
about school and district support structures that might enhance the effectiveness of 
mathematics professional development. We will outline these hypothesized support 
structures later in the next section of this chapter. 

In addition to formally testing our initial hypotheses, we will share our analysis 
of each annual round of data with the districts to provide them with feedback about 
the institutional settings in which mathematics teachers are developing and 
revising their instructional practices, and we will collaborate with them to identify 
any adjustments that might make the districts' improvement designs for middle- 
school mathematics more effective. We will then document the consequences of 
these adjustments in subsequent rounds of data collection. In addition, we will 
attempt to augment our hypotheses in the course of the repeated cycles of analysis 
and design 2 by identifying additional support structures and by specifying the 
conditions under which particular support structures are important. In doing so, we 
seek to address a pressing issue identified by Stein (2004): the proactive design of 
school and district institutional settings for mathematics teachers' ongoing 

In the remainder of this chapter, we focus on two types of conceptual tools that, 
we contend, are central to the improvement of mathematics teaching and learning 
at scale. The first is a theory of action 3 for designing schools and larger 
administrative jurisdictions as learning organizations for instructional improvement 
in mathematics. The second is an analytic approach for documenting the 
institutional setting of mathematics teaching that can produce analyses that inform 
the ongoing improvement effort. 4 


In preparing for our collaboration with the four urban school districts, we 
formulated a series of hypotheses about school and district support structures that 
we conjecture will be associated with improvement in middle-school mathematics 
teachers' instructional practices and student learning. In developing these 
hypotheses, we assumed that a school or district has adopted a research- based 
instructional programme for middle-school mathematics and that the programme 
was aligned with district standards and assessments. In addition, we assume that 
mathematics teachers have opportunities to participate in sustained professional 

2 In engaging in these repeated cycles of analysis and design, we will, in effect, attempt to conduct a 

design experiment at the level of the school and district. 

' The term theory of action was coined by Argyris and Schon ( 1 974, 1 978) and is central to most 

current perspectives on organizational learning. A theory of action establishes the rationale for an 

improvement design and consists of conjectures about both a trajectory of organizational improvement 

and the specific means of supporting the envisioned improvement process. 

4 These two types of conceptual tools serve to ground the two aspects of the design research cycle, 

namely design and analysis (e.g., Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Design-Based 

Research Collaborative, 2003). 



development that is organized around the instructional materials they use with 
students. The proposed support structures, which are summarized in Table 1, 
therefore fall outside mathematics educators' traditional focus on designing high- 
quality curricula and teacher professional development. To the extent that the 
hypotheses prove viable, they specify the types of institutional structures that a 
school or district organizational design might aim to engender as it attempts to 
improve the quality of mathematics teaching across the organization. 

As background for non-US readers, we should clarify that large school districts 
such as those with which we are collaborating have a central office whose staff are 
responsible for selecting curricula and for providing teacher professional 
development in various subject matter areas including mathematics. In this chapter, 
we use the designation district leaders to refer to members of the central office 
staff whose responsibilities focus on instruction. We speak of district mathematics 
leaders to refer to central office staff whose responsibilities focus specifically on 
the teaching and learning of mathematics. 

Table 1. 

The proposed support structures 

Primary Support 

Facilitating Support 


Teacher Networks 

Time for collaboration 
Access to expertise 

Social support for 
development of 
ambitious instructional 

Shared Instructional 


Negotiation of the 
meaning of key 
boundary objects 

Coherent instructional 
improvement effort 

Accountability Relations 
and Relations of 

Leadership content 

Effective instructional 
leadership practices 

De-tracked instructional 

Category system for 
classifying students 

Equity in students' 
learning opportunities 

Teacher Networks 

We developed our hypotheses about potential support structures by taking as our 
starting point forms of classroom instructional practice that are consistent with 
current research on mathematics learning and teaching (Kilpatrick, Martin, & 
Schifter, 2003). Teachers who have developed high quality instructional practices 
of this type attempt to achieve a significant mathematical agenda by building on 
students' current mathematical reasoning. To this end, they engage students in 
mathematically challenging tasks, maintain the level of challenge as tasks are 
enacted in the classroom (Stein & Lane, 1996; Stein, Smith, Henningsen, & Silver, 
2000), and support students' efforts to communicate their mathematical thinking in 
classroom discussions (Cobb, Boufi, McClain, & Whitenack, 1997; Hiebert et al., 



1997; Lampert, 200 1). 5 These forms of instructional practice are complex, 
demanding, uncertain, and not reducible to predictable routines (Ball & Cohen, 
1999; Lampert, 2001; McClain, 2002; Schifter, 1995; Smith, 1996). The findings 
of a number of investigations indicate that strong professional networks (see also 
Lerman & Zehetmeier, and Borba & Gadanidis, this volume) in which teachers 
participate voluntarily can be a crucial resource as they attempt to develop 
instructional practices in which they place students' reasoning at the centre of their 
instructional decision making (Cobb & McClain, 2001; Franke & Kazemi, 2001b; 
Gamoran, Secada, & Marrett, 2000; Kazemi & Franke, 2004; Little, 2002; Stein, 
Silver, & Smith, 1998). 

There is abundant evidence that the mere presence of collegial support is not by 
itself sufficient: both the focus and the depth of teachers' interactions matter. With 
regard to focus, it is clearly important that activities and exchanges in teacher 
networks centre on issues central to classroom instructional practice (Marks & 
Louis, 1997). Furthermore, the findings of Coburn and Russell's (in press) recent 
investigation indicate that the depth of interactions around classroom practice 
make a difference in terms of the support for teachers' improvement of their 
classroom practices. Coburn and Russell clarify that interactions of greater depth 
involve discerning the mathematical intent of instructional tasks and identifying 
the relative sophistication of student reasoning strategies, whereas interactions of 
less depth involve determining how to use instructional materials and mapping the 
curriculum to district or state standards. 

Teacher networks that focus on issues relevant to classroom instruction 
constitute our first hypothesized support structure. In addition, we anticipate that 
networks in which interactions of greater depth predominate will be more 
supportive social contexts for teachers' development of ambitious instructional 
practices than those in which interactions are primarily of limited depth (Franke, 
Kazemi, Shih, Biagetti, & Battey, in press; Stein et al., 1998). 

Access of Teacher Networks to Key Resources 

Mathematics teacher networks do not emerge in an institutional vacuum. Gamoran 
et al.'s (2003) analysis reveals that to remain viable, teacher networks and 
communities need access to resources. The second and third hypothesized support 
structures concern two specific types of resources that facilitate the emergence and 
development of teacher networks (see Table 1). 

Time for collaboration. The first resource is time built into the school schedule for 
mathematics teachers to collaborate. As Gamoran et al. (2003) make clear, time for 
collaboration is a necessary but not sufficient condition for the emergence of 
teacher networks. Although institutional arrangements such as teachers' schedules 
do not directly determine interactions, they can enable and constrain the social 

5 The research base for these broad recommendations is presented in a research companion volume to 
the National Council of Mathematics' (2000) Principles and Standards for School Mathematics edited 
by Kilpatrick, Martin, and Schifter (2003). 



relations that emerge between teachers (and between teachers and instructional 
leaders) (Smylie & Evans, 2006; Spillane, Reiser, & Gomez, 2006). 

Access to expertise. The second resource for supporting the emergence of teacher 
networks of sufficient depth is access to colleagues who are already relatively 
accomplished in using the adopted instructional programme to support students' 
mathematical learning. In the absence of this resource, it is difficult to envision 
how interactions within a teacher network will be of sufficient depth to support 
teachers' development of ambitious instructional practices. In this regard, Penuel, 
Frank, and Krause (2006) found that improvement in mathematics teachers' 
instructional practices was associated with access to mentors, mathematics 
coaches 6 , and colleagues who were already expert in the reform initiative. Their 
results indicate that accomplished fellow teachers and coaches can share exemplars 
of instructional practice that are tangible to their less experienced colleagues, thus 
supporting their efforts to improve their instructional practices. 

Shared Instructional Vision 

In considering additional support structures, we step back to locate teacher 
networks first within the institutional context of the school, and then within the 
context of the broader administrative jurisdiction. At the school level, it seems 
reasonable to speculate that teacher networks will be more likely to emerge and 
sustain if the vision of high quality mathematics instruction that they promote is 
consistent with the instructional vision of formal or positional school leaders. 
Research in the field of educational leadership indicates that this intuition is well 
founded. The results of a number of studies reveal that professional development, 
collaboration between teachers, and collegiality between teachers and formal 
school leaders are rarely effective unless they are tied to a shared vision of high 
quality instruction that gives them meaning and purpose (Elmore, Peterson, & 
McCarthey, 1996; Newman & Associates, 1996; Rosenholtz, 1985, 1989; Rowan, 
1990). In the case of US schools, formal school leaders might include the school 
principal, an assistant principal with responsibility for curriculum and instruction, a 
mathematics department head, and possibly a school-based mathematics coach. 
The notion of a shared instructional vision encompasses agreement on instructional 
goals and thus on what it is important for students to know and be able to do 
mathematically, 7 and on how students' development of these forms of 
mathematical knowledgeability can be effectively supported. 

6 Mathematics coaches are teachers who have been released from some or all of their instructional 
responsibilities in order to assist the mathematics teachers in a school in improving the quality of their 
instruction. Ideally, coaches should be selected on the basis of their competence as mathematics 
teachers and should receive professional development that focuses on both mathematics teaching and on 
supporting other teachers' learning. 

7 A focus on instructional goals takes us onto the slippery terrain of mathematical values (Hiebert, 
1999). It is important to note that values are not a matter of mere subjective whim or taste but are 
instead subject to justification and debate (Rorty, 1982). 



Our argument for the importance of a shared instructional vision is not restricted 
to the school but also extends to broader administrative jurisdictions. We illustrate 
this point by taking the relevant administrative jurisdiction in the US context, the 
school district, as an example. As is the case for the relevant jurisdiction in most 
countries, there are typically a number of distinct departments or units within the 
administration of large districts whose work has direct consequences for the 
teaching and learning of mathematics. For example, one unit is typically 
responsible for selecting instructional materials in various subject matter areas 
including mathematics, and for providing teacher professional development. A 
separate unit is typically responsible for hiring and providing professional 
development for school leaders. The unit responsible for assessment and evaluation 
would also appear critical given the importance of the types of data that are 
collected to assess school, teacher, and student learning. In addition, depending on 
the district, the unit responsible for special education might also be influential to 
the extent that it focuses on how mainstream instruction serves groups of students 
identified as potentially at-risk. Spillane et al.'s (2006) findings indicate that staff 
in different administrative units whose work contributes to the district's initiative 
to improve the quality of mathematics teaching and learning frequently understand 
district-wide initiatives differently. In such cases, the policies and practices of the 
various units are fragmented and often in conflict with each other. This has 
consequences both for the coherence of the district's instructional improvement 
effort and for the degree to which the institutional settings of mathematics teaching 
support teachers' ongoing improvement of their instructional practices. Our fourth 
hypothesized support structure therefore concerns the development of a shared 
instructional vision between participants in teacher networks, formal school 
leaders, and district leaders. We anticipate that mathematics teachers' improvement 
of their instructional practices will be greater in schools and broader jurisdictions 
in which a shared instructional vision consistent with current reform 
recommendations has been established. 


The development of a shared instructional vision of high quality mathematics 
instruction in a school and a broader jurisdiction such as a district is a non-trivial 
accomplishment. This becomes apparent when we note that mathematics teachers, 
principals, and district curriculum specialists, and so forth constitute distinct 
occupational groups that have different charges, engage in different forms of 
practice, and have different professional affiliations (Spillane et al., 2006). The 
fifth support structure concerns the presence of brokers who can facilitate the 
development of a shared instructional vision by bridging between perspectives and 
agendas of different role groups (see Table I). Brokers are people who participate 
at least peripherally in the activities of two or more groups, and thus have access to 
the perspectives and meanings of each group (Wenger, 1998). For example, a 
principal who participates in professional development with mathematics teachers 
might be able to act as a broker between school leaders and mathematics teachers 
in the district, thereby facilitating the alignment of perspectives on mathematics 



teaching and learning across these two groups (e.g., Wenger, 1998). Extending our 
focus beyond the school, we anticipate that brokers who can bridge between school 
and district leaders and between units of the district central office will also be 
critical in supporting the development of a shared instructional vision across the 
district. Brokers who can help bring coherence to the reform effort in a relatively 
large jurisdiction such as an urban district by grounding it in a shared instructional 
vision constitute our fifth support structure. 

Negotiating the Meaning of Key Boundary Objects 

The sixth hypothesized support structure also facilitates the development of a 
shared instructional vision (see Table 1). Mathematics teachers and instructional 
leaders use a range of tools as an integral aspect of their practices. Star and 
Griesemer (1989) call tools that are used by members of two or more groups 
boundary objects. For example, mathematics teachers and instructional leaders in 
most US schools use state mathematics standards and test scores, thereby 
constituting them as boundary objects. Tools that are produced within a school or 
district might also be constituted as boundary objects. For example, the district 
leaders in one of the districts in which we are working are developing detailed 
curriculum frameworks for middle-school mathematics teachers to use as well as a 
simplified version for school leaders. It is important to note that boundary objects 
such as state and district standards, test scores, and curriculum frameworks can be 
and are frequently used differently and come to have different meanings as 
members of different groups such as teachers and school leaders incorporate them 
into their practices (Star & Griesemer, 1989; Wenger, 1998). Boundary objects do 
not therefore carry meanings across group boundaries. However, they can serve as 
important focal points for the negotiation of meaning and thus the development of 
a shared instructional vision. The value of boundary objects in this regard stems 
from the fact that they are integral to the practices of different groups and are 
therefore directly relevant to the concerns and interests of the members of the 
groups. From the point of view of organizational design, this observation points to 
the importance of developing venues in which members of different role groups 
engage together in activities that relate directly to teaching and instructional 
leadership in mathematics. 

Our sixth hypothesis is therefore that a shared vision of high quality 
mathematics instruction will emerge more readily in schools and districts in which 
members of various groups explicitly negotiate the meaning and use of key 
boundary objects. In speaking of key boundary objects, we are referring to tools 
that are used when developing an agenda for mathematics instruction (e.g., 
curriculum frameworks) and when making mathematics teaching and learning 
visible (e.g., formative assessments, student work), as well as tools that are used 
while actually teaching. 



Accountability Relations between Teachers, School leaders, and District Leaders 

The picture that emerges from the support structures we have discussed thus far is 
that of a coherent reform effort grounded in a shared instructional vision, in which 
networks characterized by relatively deep interactions support teachers' ongoing 
learning. Although the activities of teachers as well as of school and district leaders 
are aligned in this picture, we have not specified the relationships between 
members of these different role groups. The next two potential support structures 
address this issue. 

The seventh hypothesized support structure concerns accountability relations 
between teachers, school leaders, and district leaders. At the classroom level, 
instruction that supports students' understanding of central mathematical ideas 
involves what Kazemi and Stipek (2001) term a high press for conceptual thinking. 
Kazemi and Stipek clarify that teachers maintain a high conceptual press by 1) 
holding students accountable for developing explanations that consist of a 
mathematical argument rather than simply a procedural description, 2) attempting 
to understand relations among multiple solution strategies, and 3) using errors as 
opportunities to reconceptualize a problem, explore contradictions in solutions, and 
pursue alternative strategies. Analogously, we hypothesize that the following 
accountability relations will contribute to instructional improvement: 

- Formal school instructional leaders (e.g., principals, assistant principals, 
mathematics coaches) hold mathematics teachers accountable for maintaining 
conceptual press for students and, more generally, for developing ambitious 
instructional practices. 

- District leaders hold school leaders accountable for assisting mathematics 
teachers in improving their instructional practices. 

We anticipate that the potential of these accountability relations to support 
instructional improvement will both depend on and contribute to the development 
of a shared instructional vision. In the absence of a shared vision, different school 
leaders might well hold teachers accountable to different criteria, some of which 
are at odds with the intent of the district's instructional improvement effort 
(Coburn & Russell, in press). 

Relations of Assistance between Teachers, School Leaders, and District Leaders 

Elmore (2000, 2004) argues, correctly in our view, that it is unethical to hold 
people accountable for developing particular forms of practice unless their learning 
of those practices is adequately supported. We would, for example, question a 
teacher who holds students accountable for producing mathematical arguments to 
explain their thinking but does little to support the students' development of 
mathematical argumentation. In Elmore's terms, the teacher has violated the 
principle of mutual accountability, wherein leaders are accountable to support the 
learning of those who they hold accountable. The eighth hypothesized support 
structure comprises the following relations of support and assistance: 



- Formal school instructional leaders (e.g., principals, assistant principals, 
mathematics coaches) are accountable to teachers for assisting them in 
understanding the mathematical intent of the curriculum, in maintaining 
conceptual press for students and, more generally, in developing ambitious 
instructional practices. 

- District leaders are accountable to school leaders to provide the material 
resources needed to facilitate high quality mathematics instruction, and to 
support school leaders' development as instructional leaders. 

Leadership Content Knowledge 

The ninth hypothesized support structure follows directly from the relations of 
accountability and assistance that we have outlined and concerns the leadership 
content knowledge of school and district leaders (see Table 1). Leadership content 
knowledge encompasses leaders' understanding of the mathematical intent of the 
adopted instructional materials, the challenges that teachers face in using these 
materials effectively, and the challenges in supporting teachers' reorganization of 
their instructional practices (Stein & Nelson, 2003). Ball, Bass, Hill, and 
colleagues have demonstrated convincingly that ambitious instructional practices 
involve the enactment of a specific type of mathematical knowledge that enables 
teachers to address effectively the problems, questions, and decisions that arise in 
the course of teaching (Ball & Bass, 2000; Hill & Ball, 2004; Hill, Rowan, & Ball, 
2005). Analogously, Stein and Nelson (2003) argue that effective school and 
district instructional leadership in mathematics involves the enactment of a subject- 
matter-specific type of mathematical knowledge, leadership content knowledge, 
that enables instructional leaders to recognize high-quality mathematics instruction 
when they see it, support its development, and organize the conditions for 
continuous learning among school and district staff. Stein and Nelson go on to 
argue that the leadership content knowledge that principals require to be effective 
instructional leaders in mathematics includes a relatively deep understanding of 
mathematical knowledge for teaching, of what is known about how to teach 
mathematics effectively, and of how students learn mathematics, as well as 
"knowing something about teachers-as-learners and about effective ways of 
teaching teachers" (p. 416). They extend this line of reasoning by proposing that 
district leaders who provide professional development for principals should know 
everything that principals need to know and should also have knowledge of how 
principals learn. 

We see considerable merit in Stein and Nelson's arguments about the value of 
leadership content knowledge in mathematics. However, the demands on principals 
seem overwhelming if they are to develop deep leadership content knowledge in 
all core subject matter areas including mathematics. This is particularly the case for 
principals of middle and high schools. We therefore suggest that it might be more 
productive to conceptualize this type of expertise as being distributed across formal 
and informal school leaders rather than residing exclusively with the principal. In 
other words, we suggest that the depth of leadership content knowledge that 



principals require is situational and depends in large measure on the expertise of 
others in the school. In cases where principals can capitalize on the expertise of a 
core group of relatively accomplished mathematics teachers or an effective school- 
based mathematics coach, for example, the extent of principals' leadership content 
knowledge in mathematics might not need to be particularly extensive. In such 
cases, it might suffice for principals to understand the characteristics of high 
quality instruction that hold across core subject matter areas provided they also 
understand the overall mathematical intent of the instructional programme and 
appreciate that using the programme effectively is a non-trivial accomplishment 
that requires ongoing support for an extended period of time. We speculate that 
this limited knowledge might enable principals to collaborate effectively with 
accomplished teachers and possibly school-based coaches. Stein and Nelson (2003, 
p. 444) acknowledge the viability of this approach when they observe that 

where individual administrators do not have the requisite knowledge for the 
task at hand they can count on the knowledge of others, if teams or task 
groups are composed with the recognition that such knowledge will be 
requisite and someone, or some combination of people and supportive 
materials, will need to have it. 

The ninth support structure is therefore leadership content knowledge in 
mathematics that is distributed across the principal, teachers, and the coach. This 
hypothesized support structure implies that it will be important for principal 
professional development to attend explicitly to the issue of leveraging teachers' 
and coaches' expertise effectively. 

Equity in Students ' Access to Ambitious Instructional Practices 

The student population is becoming increasingly diverse racially and ethnically in 
most industrialized countries and in a number of developing countries. An 
established research base indicates that access to ambitious instructional practices 
for students who are members of historically under-served populations (e.g., 
students of colour, students from low-income backgrounds, students who are not 
native language speakers, students with special needs) is rarely achieved (see 
Darling- Hammond, 2007). In addition, a small but growing body of research that 
suggests that ambitious instructional practices are not enough to support all 
students' mathematical learning unless they also take account of the social and 
cultural differences and needs of historically marginalized groups of students (see 
Nasir & Cobb, 2007). This work indicates the importance of professional 
development for teachers and instructional leaders in mathematics that focuses 
squarely on meeting the needs of underserved groups of students. In addition, it 
has implications for the establishment of institutional support structures that are 
likely to result in access to appropriate instructional practices for historically 
marginalized groups of students. The final two support structures that we discuss 
concern equity in students' learning opportunities. 



De-tracked instructional programme. Tracking, or the grouping of students 
according to current achievement, often prevails in schools that serve students 
from marginalized groups. However, current research indicates that "tracking does 
not substantially benefit high achievers and tends to put low achievers at a serious 
disadvantage" (Darling-Hammond, 2007, p. 324; see also Gamoran, Nystrand, 
Berends, & LePore, 1995; Horn, 2007; Oakes, Wells, Jones, & Datnow, 1997). 
The tenth support structure is therefore a rigorously de-tracked instructional 
programme in mathematics. 

Category system for classifying students. The final support structure concerns the 
categories of mathematics students that are integral to teachers' and instructional 
leaders' practices. Horn's (2007) analysis of the contrasting systems for classifying 
students constructed by the mathematics teachers in two US high schools is 
relevant in this regard because it indicates that these classification systems were 
related to the two groups of teachers' views about whether mathematics should be 
tracked (see Table 1). Significantly, Horn's analysis also indicates that the 
contrasting classification systems also reflected differing views of mathematics as 
a school subject. The teachers in one of the schools differentiated between formal 
and informal solution methods, and viewed the latter as illegitimate. They also 
took a sequential view of school mathematics and assumed that students had to 
first master prior topics if they were to make adequate progress. This conception of 
school mathematics was reflected in the teachers' classification of students as more 
or less motivated to master mathematical formalisms, and as faster and slower in 
doing so. The teachers' classification of students in terms of stable levels of 
motivation and ability grounded their perceived need for separate mathematics 
courses for different types of students. 

In sharp contrast, the mathematics teachers at the second school that Horn 
(2007) studied tended to take a non-sequential view of school mathematics and 
conceptualized it as a web of ideas rather than an accumulation of formal 
procedures. These teachers also rejected the categorization of students as fast or 
slow because it emphasized task completion at the expense of considering multiple 
strategies. In addition, the teachers in this school viewed it as their responsibility to 
support students' engagement both by selecting appropriate tasks and by 
influencing students' learning agendas. Thus, these teachers addressed the 
challenge of teaching mathematics to all their students in the context of a 
rigorously de-tracked mathematics programme by focusing primarily on their 
instructional practices rather than on perceived mismatches between students and 
the curriculum. In doing so, they constructed categories for classifying students 
that characterized them in relation to their current instructional practices rather 
than in terms of stable traits. Building on Horn's analysis, the eleventh support 
structure is a category system that classifies students in relation to current 
instructional practices rather than in terms of seemingly stable traits. 




We developed the proposed support structures summarized in Table 1 by mapping 
backwards from the classroom and, in particular, from a research-based view of 
high quality mathematics instruction. In doing so, we have limited our focus to the 
establishment of institutional settings that support school and district staffs 
ongoing improvement of their practices. This backward mapping process could be 
extended to develop conjectures that are directly related to the traditional concerns 
of policy researchers. For example, several of the hypothesized support structures 
involve conjectures about the role of mathematics coaches and school leaders. 
These conjectures have implications for district hiring and retention policies. In 
addition, the hypotheses imply that the allocation of frequently scarce material 
resources should be weighted towards what Elmore (2006) terms the bottom of the 
system (see also Gamoran et al., 2003). As the notion of distributed leadership is 
currently fashionable, 8 it is worth noting that the hypotheses do not treat the 
distribution of instructional leadership as a necessary good. In the absence of a 
common discourse about mathematics, learning, and teaching, the distribution of 
leadership can result in a lack of coordination and alignment (Elmore, 2000). As 
Elmore (2006) observes, effective schools and districts do not merely distribute 
leadership. They also support people 's development of leadership capabilities, in 
part by structuring settings in which they learn and enact leadership. As the 
proposed support structures indicate, important outcomes of an initiative to 
improve the quality of mathematics learning and teaching include "the system 
capacity developed to sustain, extend, and deepen a successful initiative" (Elmore, 
2006, p. 219). 


The hypothesized support structures that we have discussed constitute a theory of 
action for designing schools and larger administrative jurisdictions such as school 
districts as learning organizations for instructional improvement in mathematics. 
We now consider a second conceptual tool that is central to the improvement of 
mathematics teaching and learning at scale, an analytic approach for documenting 
the institutional setting of mathematics teaching. In addition to formally testing our 
hypotheses about potential support structures, we will share our analysis of the data 
collected each year with the four districts and collaborate with them to identify any 
adjustments that might make the districts' improvement designs for middle-school 
mathematics more effective. To accomplish this, we require an analytic approach 
for documenting the institutional setting of mathematics teaching that can feed 
back to inform the districts' ongoing improvement efforts. 

Spillane and colleagues (Spillane, 2005; Spillane, Halverson, & Diamond, 2001, 2004) proposed 
distributed leadership as an analytic perspective that focuses on how the functions of leadership are 
accomplished rather than on the characteristics and actions of individual positional leaders. However, as 
so often happens in education, the basic tenets of this analytic approach have been translated into 
prescriptions for practitioners' actions. In our view, this is a fundamental category error that, if past 
experience is any guide, might well have unfortunate consequences (e.g., Cobb, 1994, 2002). 



The analytic approach that we will take makes a fundamental distinction 
between schools and districts viewed as designed organizations and as lived 
organizations. A school or district viewed as a designed organization consists of 
formally designated roles and divisions of labour together with official policies, 
procedures, routines, management systems, and the like. Wenger (1998) uses the 
term designed organization to indicate that its various elements were designed to 
carry out specific tasks or to perform particular functions. In contrast, a school or 
school district viewed as a lived organization comprises the groups within which 
work is actually accomplished together with the interconnections between them. 
As Brown and Duguid (1991, 2000) clarify, people frequently adjust prescribed 
organizational routines and procedures to the exigencies of their circumstances 
(see also Kawatoko, 2000; Ueno, 2000; Wenger, 1998). In doing so, they often 
develop collaborative relationships that do not correspond to formally appointed 
groups, committees, task forces, and teams (e.g., Krainer, 2003). Instead, the 
groups within which work is actually organized are sometimes non-canonical and 
not officially recognized. These non-canonical groups are important elements of a 
school or district viewed as a lived organization. 

Given the goals of our research, we find it essential to document the districts in 
which we are working as both designed organizations and as lived organizations. 
One of our first steps has been to document the districts as designed organizations 
by interviewing district leaders about their plans or designs for supporting the 
improvement of mathematics teaching and learning. In analysing these interviews, 
we have teased out the suppositions and assumptions and have framed them as 
testable conjectures. The process of testing these conjectures requires that we 
document how the districts' improvement designs are playing out in practice, 
thereby documenting the schools and districts in which we are working as lived 

Methodologically, we will use what Hornby and Symon (1994) and Spillane 
(2000) refer to as a snowballing strategy and Talbert and McLaughlin (1999) term 
a bottom-up strategy to identify groups within the schools and districts whose 
agendas are concerned with the teaching and learning of mathematics. The first 
step in this process involves conducting audio-recorded semi-structured interviews 
with the participating 30 middle-school mathematics teachers in each district to 
identify people within the district who influence how the teachers teach 
mathematics in some significant way. The issues that we will address in these 
interviews include the professional development activities in which the teachers 
have participated, their understanding of the district's policies for mathematics 
instruction, the people to whom they are accountable, their informal professional 
networks, and the official sources of assistance on which they draw. 

The second step in this bottom-up or snowballing process involves interviewing 
the formal and informal instructional leaders identified in the teacher interviews as 
influencing their classroom practices. The purpose of these interviews is to 
understand formal and informal leaders' agendas as they relate to mathematics 
instruction and the means by which they attempt to achieve those agendas. We will 
then continue this snowballing process by interviewing people identified in the 



second round of interviews as influencing instruction and instructional leadership 
in the district. In terms familiar to policy researchers, this bottom-up methodology 
focuses squarely on the activity of what Weatherley and Lipsky (1977) term street- 
level bureaucrats whose roles in interpreting and responding to district efforts to 
improve mathematics instruction are as important as those of district leaders who 
designed the improvement initiative. The methodology therefore operational izes 
the view that what ultimately matters is how district initiatives are enacted in 
schools and classrooms (e.g., McLaughlin, 2006). 

In addition to identifying the groups in which the work of instructional 
improvement is accomplished and documenting aspects of each group's practices, 
our analysis of the schools and districts as lived organizations will also involve 
documenting the interconnections between the groups. To do so, we will focus on 
three types of interconnections, two of which we introduced when describing 
potential support structures. Interconnections of the first type are constituted by the 
activities of brokers who are at least peripheral members of two or more groups. 
As we noted, brokers can bridge between the perspectives of different groups, 
thereby facilitating the alignment of their agendas. As our hypotheses indicate, our 
analysis of brokers will be relatively comprehensive and will seek to clarify 
whether there are brokers between various groups in the school (e.g., mathematics 
teachers and school leaders), between school leaders and district leaders, and 
between key units of the district central office. Boundary objects that members of 
two or more groups use routinely as integral aspects of their practices constitute 
interconnections of the second type. As we have noted, there is the very real 
possibility that members of different groups will used boundary objects differently 
and imbue them with different meanings (Wenger, 1998). Our analysis will 
therefore seek to identify boundary objects and to document whether members of 
different groups used them in compatible ways. 

The third type of interconnection is constituted by boundary encounters in 
which members of two or more groups engage in activities together as a routine 
part of their respective practices. Three of the hypothesized support structures 
focus explicitly on boundary encounters: the explicit negotiation of the meaning of 
boundary objects, relations of accountability, and relations of assistance. In 
addition to documenting the frequency of boundary encounters between members 
of different groups, our analysis will focus on the nature of their interactions. 

A recent finding reported by Cob urn and Russell (in press) indicates the 
importance of pushing for this level of detail. They studied the implementation of 
elementary mathematics curricula designed to support ambitious instruction in two 
school districts. As part of their instructional improvement efforts, both districts 
hired and provided professional development for a cadre of school-based 
mathematics coaches (see also Nickerson, this volume). Coburn and Russell found 
that there were significant differences in the depth of the interactions between the 
coaches and the professional development facilitators in the two districts. In the 
first district, interactions were relatively deep and focused on issues such as 
discerning the mathematical intent of instructional tasks and on identifying and 
building on student reasoning strategies. In the second district, interactions were 



typically of limited depth and focused primarily on how to use instructional 
materials and on mapping the curriculum to district or state standards. Coburn and 
Russell also documented the nature of interactions between coaches and teachers in 
the two districts. They found that teacher-coach interactions increased in depth to a 
far greater extent in the first district than in the second district. In addition, 
interactions between teachers when a coach was not present also increased in depth 
in the first but not the second district. In other words, the contrasting routines of 
interaction in coach professional development sessions became important features 
of interactions in teacher networks in the two districts. 

In our view, Coburn and Russell's analysis represents a significant advance in 
research on instructional improvement at scale. To this point, policy researchers 
have tended to frame social networks as conduits for information about 
instructional and instructional leadership practices. However, research in 
mathematics education makes it abundantly clear that information about ambitious 
instructional practices is, by itself, insufficient to support teachers' development of 
this form of practice. Coburn and Russell's analysis focuses more broadly on 
interactions across groups as well as within social networks, and highlights the 
importance of co-participation in collective activities. In addition, their findings 
demonstrate that the depth of co-participation matters. Their analysis therefore 
establishes a valuable point of contact between research on policy implementation 
and research on mathematics teachers' learning. This latter body of work 
documents that teachers' co-participation in activities of sufficient depth with an 
accomplished colleague or instructional leader is a critical source of support for 
teachers' development of ambitious practices (e.g., Borko, 2004; Fennema et al., 
1996; Franke & Kazemi, 2001a; Goldsmith & Shifter, 1997; Kazemi & Franke, 
2004; Wilson & Berne, 1999). We anticipate that Coburn and Russell's (in press) 
notion of routines of interaction will prove to be a useful analytic tool as we seek 
to understand whether the nature of the boundary encounters in which school and 
district staff engage in activities together influences how they subsequently interact 
with others in different settings. 


In the approach that we have outlined, the analysis of a school or district as a lived 
organization involves identifying the groups in which the work of instructional 
improvement is actually accomplished and documenting interconnections between 
these groups. An analysis of the lived organization therefore focuses on what 
people actually do and the consequences for teachers' instructional practices and 
students' mathematical learning. In contrast, an analysis of a school or district as a 
designed organization involves documenting the school or district plan or design 
for supporting instructional improvement in mathematics. This design specifies 
organizational units and positional roles as well as organizational routines, and 
involves conjectures about how the enactment of the design will result in the 
improvement of teachers' instructional practices and student learning. An analysis 
of the designed organization documents both this design and the tools and 
activities that will be employed to realize the design by enabling people to improve 



their practices. In giving feedback to the four collaborating districts to inform their 
improvement efforts, we will necessarily draw on our analyses of the districts as 
both designed and lived organizations. 

To develop this feedback, we will identify gaps between the districts' designs 
for instructional improvement and the ways in which those designs are actually 
playing out in practice by comparing our analyses of each district as a designed 
organization and as a lived organization. This approach will enable us to 
differentiate cases in which a theory of action proposed by a district is not enacted 
in practice from cases in which the enactment of the theory of action does not lead 
to the anticipated improvements in the quality of teachers' instructional practices 
(Supovitz & Weathers, 2004). As an illustration, one of the districts with which we 
are collaborating is investing some of its limited resources in mathematics coaches 
with half-time release from teaching for each middle school. The district's theory 
of action specifies that the coaches' primary responsibilities are to facilitate teacher 
collaboration and to support individual teachers' learning by co-teaching with them 
and by observing their instruction and providing constructive feedback. Suppose 
that the district's investment in mathematics coaches does not result in a noticeable 
improvement in teachers' instructional practices. It could be the case that the 
theory of action of the district has not been enacted. For example, the coaches 
might be tutoring individual students or preparing instructional materials for the 
mathematics teachers in their schools rather than working with teachers in their 
classrooms. In attempting to understand why this is occurring, we would initially 
focus on coaches' and school leaders' understanding of the coaches' role in 
supporting teachers' improvement of their instructional practices. Alternatively, it 
could be the case that the coaches are working with teachers in their classrooms, 
but their efforts to support instructional improvement are not effective. In this case, 
we would initially seek to understand how, specifically, the coaches are attempting 
to support teachers' learning and would take account of the process by which the 
coaches were selected and the quality of the professional development in which 
they participated. 

As this illustration indicates, our goal when giving feedback is not merely to 
assess whether the district's design is being implemented with fidelity, although 
our analysis will necessarily address this issue. We also seek to understand why the 
district's theory of action is playing out in a particular way in practice by taking 
seriously the perspectives and practices of street-level bureaucrats such as teachers, 
coaches, and school leaders. In doing so, we will draw on both an analysis of the 
district design as a potential resource for action and an analysis of the district as a 
lived organization that foregrounds people's agency as they develop their practices 
within the context of others' institutionally situated actions (e.g., Feldman & 
Pentland, 2003). 


In this chapter, we have focused on the question of how mathematics education 
research might contribute to the improvement of mathematics teaching and 
learning at scale. We addressed this question by first clarifying the value of 



viewing mathematics teachers' instructional practices as situated in the institutional 
settings of the schools and districts in which they work. Against this background, 
we presented a series of hypotheses about school and district structures that might 
support teachers' ongoing improvement of their classroom practices. We then went 
on to outline an analytic approach for documenting the institutional settings of 
mathematics teaching established in particular schools and districts that can feed 
back to inform the instructional improvement effort. 

We conclude this chapter by returning to the relation between research in 
educational policy and leadership and in mathematics education. To this point, 
researchers in these fields have conducted largely independent lines of work on the 
improvement of teaching and learning (e.g., EngestrSm, 1998; Franke, Carpenter, 
Levi, & Fennema, 2001). Research in educational policy and leadership tends to 
focus on the designed structural features of schools and how changes in these 
structures can result in changes in classroom instructional practices. In contrast, 
research in mathematics education tends to focus on the role of curriculum and 
professional development in supporting teachers' improvement of their 
instructional practices and their views of themselves as learners. In this chapter, we 
have argued that mathematics education research that seeks to contribute to the 
improvement of teaching and learning at scale will have to transcend this 
dichotomy by drawing on analyses of schools and districts viewed both as 
designed organizations and as lived organizations. In the interventionist genre of 
research that we favour, organizational design is at the service of large-scale 
improvement in the quality of teachers' instructional practices. In research of this 
type, the attempt to contribute to improvement efforts in particular schools and 
administrative jurisdictions constitutes the context for the generation of useful 
knowledge about the relations between the institutional settings in which teachers' 
work, the instructional practices they develop in those settings, and their students' 
mathematical learning. This genre of research therefore reflects de Corte, Greer, 
and Verschaffel's (1996) adage that if you want to understand something try to 
change it, and if you want to change something try to understand it. 


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Paul Cobb 

Vanderbilt University 

Thomas Smith 
Vanderbilt University 







The chapter considers four examples of large-scale projects involving national 
reform initiatives in mathematics drawn from four continents — Europe, North 
America, Australia and Asia. Poor student performance on international and/or 
national assessment programmes was, in part, a catalyst for each programme. The 
underlying issue driving each of these studies was the perceived importance of 
improved student learning outcomes in mathematics and science in these countries. 
All the projects focus on initiating purposeful pedagogic change through involving 
teachers in rich professional learning experiences. The primary purpose of this 
chapter is two-fold First, a brief description is provided of each project in order to 
give an insight into different countries ' efforts to improve teaching and learning at 
scale. Second an analysis and discussion of common features are undertaken 
leading to lessons learned 


All over the world, countries face challenges in terms of supporting students to 
achieve their potential in the important area of mathematics. As we move into the 
21 st century, we find new meaning to calls about the importance of mathematics 
knowledge and know-how to economic growth. 

Glenn (2000, p. 7) writing for a US readership, identified four important and 
enduring reasons for the need for students to achieve to their potential in 
mathematics and science. These were: 

- The rapid pace of change in both the increasingly interdependent global 
economy and in the American workplace demands widespread mathematics- 
and science-related knowledge and abilities; 

- Our citizens need both mathematics and science for their everyday decision- 

- Mathematics and science are inextricably linked to the nation's security 
interests; and 

- The deeper, intrinsic value of mathematical and scientific knowledge shape and 
define our common life, history, and culture. Mathematics and science are 
primary sources of lifelong learning and the progress of our civilization. 

These or similar sentiments have prompted education authorities around the 
world to look at better ways of (i) increasing the mathematics skills of students, (ii) 

K. Krainer and T. Wood (eds.), Participants in Mathematics Teacher Education, 255-280. 
© 2005 Sense Publishers. All rights reserved. 


developing more effective programmes to support teachers meet the demands of 
modern classrooms, and (iii) increasing the attraction and retention of qualified 
mathematics teachers by improving the education of prospective teachers as well as 
addressing the professional needs of practising teachers. 

Any student who experiences ongoing failure in school faces a myriad of 
difficulties in achieving long-term employment, and useful and fulfilling 
occupations. Those who exhibit consistent weaknesses in basic skills in 
mathematics are particularly vulnerable. Test data provide a compelling case for 
the need to develop programmes and approaches that improve mathematics 
outcomes for all students and particularly for those who are performing at or below 
the international or national benchmarks. Equally important is the need of 
mathematics instruction to address affective aspects of learning. It is a great 
challenge to raise all students' interest in and joy of mathematics, to let them 
experience the benefit and beauty of this subject and to decrease their fears and 
feelings of failing, in particular taking into account gender and diversity issues. 
Cognitive and affective factors are closely interconnected. 

International data point out that the life chances of students to acquire 
mathematics competencies are not uniform across a nation. Some students 
approach education (and the learning of mathematics) with advantages that are not 
found throughout the population. In particular, those who live in poverty, in rural 
locations or belong to at-risk minorities often have less chance of success than 
others. Hence, driving many reform initiatives is the principle of equity of 
educational opportunity regardless of current life opportunities as well as an 
expectation of achievement regardless of the current ability level. 

The central question for most nations is where best to direct efforts so that 
meaningful and sustainable change can be managed in a cost-effective way. While 
the problems each nation faces are complex and unique, much can be gained by 
analysing different approaches. The large-scale projects described in this text, 
highlight how four countries have, in part, attempted to address education issues. 
While their programmes are sophisticated in design and implementation, they have 
a central, common theme, namely, they are directed at the professional learning of 
prospective and practising teachers. 

This focus on teachers' learning is not arbitrary: it is evident that reform 
programmes must access students through their teachers. However, how large is 
teachers' impact on students' learning? Over the past few years, a consistent theme 
has begun to emerge concerning the variance identified in the analysis of student 
learning over many large-scale projects. Identified factors that contribute to major 
sources of variation in student performance (Hattie, 2003) include student (50%), 
home (5-10%), schools (5-10%), principals (mainly accounted for in schools), peer 
effects (5-10%), and teachers (30%). This research implies that genuine 
improvement in learning can be achieved by improving the beliefs, emotions, 
knowledge, and practice of teachers. The picture is consistent; it is what 
mathematics teachers believe, feel, know, and do, that is a powerful determinant in 
student learning. 



In the fol lowing, four large-scale projects involving significant national reform 
initiatives from four continents are described. Given the complexity and authors' 
deeper knowledge of their own initiatives in Austria and Australia, a slightly more 
extensive focus is placed on these two cases. The four countries vary greatly in size 
(and distances to be travelled), spread of the population, importance of teacher 
shortages, and the educational disadvantage experienced by minorities. The first 
two projects can be traced back to the results of student assessments in Austria 
(Europe) and the state of Ohio (US, North America). The remaining two initiatives 
address in different ways the under-achievement of students in rural and regional 
areas in comparison to their metropolitan peers. One is from Australia and the 
other is from South Korea (Asia). 

Hence, as we consider the different initiatives outlined we identify two 
underpinning tenets. First, more than at any time in our history, more citizens than 
ever need to achieve their mathematically potential if they are to have active and 
fulfilling occupations. Second, if we are to bring about sustainable change then 
money and effort needs to be directed to encouraging and supporting quality 

In order to facilitate comparison across the four projects, a common structure 
has been employed in the next section. Each description considers: 

- Impulse for the initiative and challenge 

- Goals and intervention strategy 

- Implementation and communication 

- Evaluation and impact 

- Challenges and further steps 


Austria participated in all three cohorts (primary, middle and high school) of the 
1995 TIMSS achievement study. Whereas the results for the primary and the 
middle school were rather promising, the results of the Austrian high school 
students (grades 9 to 12 or 13), particularly with regard to the TIMSS advanced 
mathematics and physics achievement test, shocked the public (Mullis et al., 
1998). These data were a catalyst for the government and education community to 
re-evaluate the status of mathematics and science education and saw the 
responsible federal ministry launch the IMST - Innovations in Mathematics and 
Science Teaching - research project (1998-1999). The purpose of this nation-wide 
programme was to address an identified complex picture of diverse problematic 
influences on the status and quality of mathematics and science teaching. 

Impulse for the Initiative and Challenge 

In addition to the disappointing performance of Austria's high school students on 
TIMSS, Austria was among those nations with the highest achievement differences 
between boys and girls. Also, students showed poor results with regard to items 



that referred to higher levels of thinking, and less than a third of Austrian students 
felt that they were involved in reasoning tasks in most or every mathematics 

In seeking to explain these results and findings, Austrian researchers were 
convinced that there were manifold causes. For example, the answers to a written 
questionnaire by Austrian teachers, teacher educators and representatives of the 
education authorities showed that teachers were predominantly seen as dedicated 
and as having a lot of pedagogical and didactic autonomy. On the one hand, there 
were many creative initiatives being carried out by individuals, groups or 
institutions; on the other hand, many of these initiatives were carried out in 
isolation, and a networking structure was missing. 

Mathematics education and related research was seen as poorly anchored at 
Austrian teacher education institutions. Subject experts dominate university 
teacher education, other teacher education institutions show a lack of research in 
mathematics education; the collaboration with educational sciences and schools is 
- with exception of a few cases - underdeveloped. A competence centre like those 
found in many other countries was not existent. Also, the overall structure 
(including two institutions for the education of prospective teachers that are mostly 
unconnected, a variety of different kinds of schools with corresponding 
administrative bodies in the ministry and in the institutions for the education of 
practising teachers, etc.) showed a picture of a "fragmentary educational system" 
of lone fighters with a high level of (individual) autonomy and action, however, 
there was little evidence of reflection and networking (Krainer, 2001). 

Goals and Intervention Strategy 

The analyses led to the four year project IMST 2 (2000-2004) - now called 
Innovations in Mathematics, Science and Technology Teaching: the addition of 
"Technology" in the project title was to express the fundamental importance of 
technologies for mathematics and science teaching. The project (Krainer, Dorfler, 
Jungwirth, Kiihnelt, Rauch, & Stern, 2002) focused solely on the upper secondary 
school level and involved the subjects, biology, chemistry, mathematics and 
physics. IMST 2 was financed by the responsible federal ministry and the Austrian 
Council of Research and Technology Development. In order to take systemic steps 
to overcome the "fragmentary educational system", the approach of a "learning 
system" (Krainer, 2005a) was taken. It adopted enhanced reflection and 
networking as the basic intervention strategy. The theoretical framework builds on 
the ideas of action research (Altrichter, Posch, & Somekh, 1993), constructivism 
(von Glasersfeld, 1991) and systemic approaches to educational change and 
system theory (Fullan, 1993; Willke, 1999). 

Besides stressing the dimensions of reflection and networking, "innovation" and 
"work with teams" were two additional features. Innovations were not regarded as 
singular events that replace an ineffective practice but as continuous processes 
leading to a natural further development of practice. Teachers and schools defined 
their own starting point for innovations and were individually supported by 



researchers and expert teachers. The IMST 2 intervention built on teachers' 
strengths and aimed at making their work visible (e.g., by publishing teachers' 
reports on the website). Thus teachers and schools retained ownership of their 
innovations. Another important feature of IMST 2 was the emphasis on supporting 
teams of teachers from a school. 
The two major tasks of IMST 2 were 

- The initiation, promotion, dissemination, networking and analysis of 
innovations in schools (and to some extent also in teacher education at 
university); and 

- Recommendations for a support system for the quality development of 
mathematics, science and technology teaching. 

The second task led to a plan for a sustainable support system (Krainer, 2005b). 
Consequently, IMST 2 was followed by the project IMST3 (2004-2007) which 
included all secondary schools and later by the project IMST3 Plus (2007-2009) 
which broadened the support of schools to the primary level. 

In IMST3 and IMST3 Plus about three times more schools were supported and 
also the participation in regional networks was fostered. It was intended to build 
up a network of practitioners, researchers and administrative staff that to help to 
support the schools. Another task was to contribute to the implementation of a 
better infrastructure for mathematics and science education. The improved 
infrastructure was considered a basis and precondition for implementing a 
sustainable network of persons and institutions. Therefore, for example, the IMST 
project team designed a plan on how to establish competence centres for 
mathematics, science and technology education. 

Implementation and Communication 

The operative implementation of most parts of IMST has been entrusted to the 
Institute of Instructional and School Development at the University of Klagenfiirt. 
Though having this university institute as a key player in the whole process, the 
whole project was understood as an initiative, and influenced by a wide network of 
people and institutions in order to improve mathematics and science teaching and 
learning in Austria. 

In the years 2000-2004, IMST supported about 50 innovation projects at 
Austrian upper secondary schools (and partially at other organisations, e.g., 
teacher education institutions) in each school-year. The participation was 
voluntary and gave teachers and schools a choice among four priority programmes 
(Basic education; School development; Teaching and learning processes; Practice- 
oriented research: Students' independent learning) according to the challenges 
sifted out in the above mentioned research project. In general, teachers in all four 
priority programmes - and also later in a specific programme on gender sensitivity 
and gender mainstreaming - were supported by staff members of IMST. The 
priority programmes can be regarded as small professional communities that not 
only supported each participant to proceed with his or her own project but also 
generated a deeper understanding of the critical reflection of one's own teaching, 



of formulating research questions, of looking for evidence based on viable data, 
and on methods that help to gather that data. 

Since 2004, the direct support of about 1 50 innovative projects is organized 
within an IMST fund. Whereas the fund aims primarily at reaching experienced 
teachers who are able to disseminate their experiences and results to other teachers 
(at their school, in their district or nation-wide), the formation of regional networks 
in each federal state aims at reaching a greater number of teachers. In addition, a 
gender network and a project "examination culture" have been established in order 
to offer advice and professional development activities for teachers. All these four 
measures of IMST3 (fund, regional networks, gender network and examination 
culture) were continued in the phase of IMST3 Plus. Step by step these measures 
are opened to teachers at the primary level. 

Throughout all phases of IMST, the project is accompanied by a website, an 
annual conference and a quarterly newsletter. 

Evaluation and Impact 

Evaluation was an integral part of IMST since its start (Krainer, 2007). The self- 
evaluation comprised forms of a process-oriented evaluation (generating steering 
knowledge for the project management and the project teams), an outcome- 
oriented evaluation (working out the impact of the project at different levels of the 
educational system), and a knowledge-oriented evaluation (generating new 
theoretical and practical knowledge about the interconnection between the 
project's interventions and its impact). In many cases, this self-evaluation included 
data gathering and feedback by external experts. In addition, several independent 
evaluations were commissioned looking at the impact of the project or parts of it. 
For example, three international experts evaluated IMST 2 (2000-2004) and IMST3 
(2004-2006) at the end of these periods of the project and wrote corresponding 
reports. Also parts of IMST3 (e.g., the priority programmes or the networks) were 
externally evaluated. 

The self-evaluation of IMST was done at different levels, in particular at the 
school, classroom and individual teachers' level, and at the Austrian educational 
system level. The teachers used different forms of action research methods. In 
some cases, they were supported by their mentoring teams by means of external 
evaluation (interviews, questionnaires, analyses of videos). Overall, the reports 
indicated significant gains of students' and teachers' affective and cognitive 

Specht (2004) reported that IMST 2 is seen as an important, useful and effective 
support for instructional and school development in mathematics and science. He 
(p. 51) identified concrete changes in teachers' orientations to actions, in particular 
concerning their readiness for innovations in teaching, their increased ability to 
reflect and self-evaluate, their higher care in choosing teaching contents and their 
more intensive collaboration with colleagues. 

With the start of IMST3 in 2004, efforts to investigate the impact of the 
programme were increased. In particular, student questionnaires and more 



systematic analyses of teachers' reports were introduced. Answers to the teacher 
questionnaire demonstrated a high level of satisfaction with the programme. 
Teachers were highly motivated even though the additional workload of the 
programme could be substantial. IMST students were generally more interested in 
their subjects (in comparison to students in the Austrian PISA sample), and 
reported less anxiety. However, this may be due to their teachers, who show a high 
level of motivation already before they entered the IMST programmes. 

A longitudinal study (Muller, Andreitz, Hanfstingl, & Krainer, 2007) based on 
the self-determination theory of Deci and Ryan (2002) was performed in all 
classes taking part. The study showed that teachers who experience support from 
their colleagues and principal assess their students as being more motivated. Their 
students felt more intrinsically motivated than students from less supported 
teachers. However, if teachers felt pressure from colleagues and the principal, 
teachers' and students' intrinsic motivation decreased. Overall, the study showed 
that teachers trying to improve their practice should not be isolated. 

An external evaluation of the regional networks of IMST 2 , on the basis of a 
questionnaire directed at principals and superintendents (Heffeter, 2006, p. 47), 
found that this particular measure promotes the communication and change of 
experiences among teachers. The study also suggested that the IMST process has 
the potential to break up thinking patterns that have become entrenched in the 
Austrian educational system. 

The external evaluations of IMST 2 and IMST3 by international experts 
(Prenzel, Schratz, & Messner, 2007) regarded the project as a national and 
international remarkable and successful development programme. Suggestions 
focused to a large extent on an increasing emphasis on generating new scientific 
knowledge that coincides with the main strategy of IMST. 

Challenges and Further Steps 

In the remaining time of IMST3 Plus (2007-2009), four major challenges are to be 
taken into consideration. Firstly, the work with primary schools has to be started. 
This is connected with intensive collaborations with the new established teacher 
education institutions (Padagogische Hochschulen) in Austria. Secondly, the 
recent formation of IMST networks at the district level (e.g., the VIAMATH 
project) generates new questions of adequate support and evaluation. Here, a good 
interplay between the IMST fund and the regional networks seems necessary as 
well as collaborations with the regional school boards and the teacher education 
institutions. Thirdly, the ongoing discussion on the plan for the time after 2009 has 
to be finalized. Fourthly, steps towards improving the opportunities for research in 
mathematics education at the primary school level have to be taken. It should be 
mentioned that Austria has so far no professor for mathematics education at the 
primary school level that means that research in that area is rather underdeveloped. 
Recently, IMST promotes the establishment of regional centres for mathematics 
and science education where teacher educators and researchers working at the 
primary and the secondary level are expected to collaborate. 





Traditionally, education in the US was a matter for local communities to control 
and fund. The publication of "A Nation at Risk" (NCEE, 1983) initiated a US- 
wide desire to move schools towards universal standards for accountability in 
mathematics and science teaching and learning. The standards-movement led to 
the fact that today both state and federal governments play a much larger role in 
US education. While federal documents contained recommendations for improving 
teaching practices and student learning for all, the specific means for achieving 
theses goals were left to individual states. As part of this pattern, Ohio established 
a model curriculum for science, established outcomes to be measured, and recently 
developed content standards. Ohio was one of the first states in the US to receive 
federal funds under the State Systemic Initiative of the National Science 
Foundation (NSF). Ohio's Systemic Initiative, Project Discovery, was initially 
funded from 1991-1996 (more info can be found at The following description is based 
largely on Wagner and Meiring (2004) and Beeth, McCollum, and Tafel 

Impulse for the Initiative and Challenge 

In addition to the federal concerns about mathematics and science learning, there 
was also a specific concern in Ohio: despite the fact that this state ranked high 
(sixth) in science and technology based industry in the United States, Ohio's 
schools ranked only in the mid-twenties in those subjects. Public media reports on 
student learning brought attention to the need for reform of educational practices. 

Ohio's Department of Education issued standards that asked for a more problem 
solving and inquiry based approach to classroom instruction. The Discovery 
Project Summer Institutes offered teachers a place to work on their professional 

Goals and Intervention Strategy 

The overall goal of Discovery was to improve mathematical and scientific 
knowledge of middle school students, and to achieve equity between students of 
different ethnic groups as well as gender equity through inquiry based instruction. 
At the time of conception, national standards were not yet developed, thus 
Discovery preceded the later move to achieve national coherence in the conception 
of mathematics and science teaching. 

Unlike other approaches to educational reform, the Discovery project did not 
simply standardise a curriculum, but addressed the methods in which science and 
mathematics are taught. The project's principal goal was to educate the educators 
and produce a cadre of teachers capable of implementing inquiry based instruction 



in the classroom. In particular, Discovery focused on increasing teachers' content 

The Discovery project went on to emphasize partnerships between teachers and 
parents in mathematics and science education and incorporated methods to address 
learning gaps between demographic groups. 

Implementation and Communication 

Discovery installed six-week professional development courses for in-service 
teachers, which taught research based inquiry methods for individual middle- 
school teachers. These courses were taught by university researchers and science 
educators as well as master teachers throughout Ohio. The inquiry based method 
focused on interaction and communication between students, discussion of 
alternative methods of problem solving, supporting claims with data and less 
emphasis on traditional rote memorization. Participants observed and discussed 
each other's teaching of classes targeted at inquiry learning. After the course, 
participants met several times during the following year working on a portfolio and 
to exchange their experiences. 

After the initial funding period, Discovery institutes have been continuing to 
offer courses, albeit to a lesser extent. The initiative expanded to include teachers 
at all grade levels and to include some site-based and other administrators, as well. 

Evaluation and Impact 

Evaluation of the programme occurred at three levels. Firstly, the scientific and 
mathematical knowledge of students participating in the Discovery project was 
compared with that of students not participating in the programme. Students in the 
programme fared an average of 7% better on standardized tests. Secondly, gender 
and ethnicity based performance gaps were assessed over time; and the programme 
was shown to decrease the gap in achievement for these groups of learners. 

Finally, passing rates of students for different schools were compared. This 
comparison showed improvements in passing grades for schools with higher 
percentages of Discovery trained teachers. An outgrowth of Discovery has been the 
establishment of multiple professional development initiatives and networks of K- 
16 institutions through a variety of state and national funding mechanisms. 

Challenges and Further Steps 

The activities leading to the most remarkable improvements in the study were 
considered too expensive to implement at a state-wide level. To mitigate some of 
the programme costs the state focused its efforts on the 9th and 10th grades, and 
distributed Discovery training materials to schools capable of implementing the 
programme. In 1995, the state of Ohio initiated the SUSTAIN project to maintain 
the results already achieved by the federally funded project Discovery. 



SUSTAIN's primary focus is in creating a partnership between state universities 
and state schools by providing inquiry based education methods to its education 
students. SUSTAIN was also designed to foster collaboration among Ohio's higher 
education institutions and with public school districts through regional professional 
development centres named Centres of Excellence in Mathematics and Science. 

At present, considerable investment is directed towards technology based, K-16, 
educational and professional development support structures delivered through 
both virtual and clinical learning experiences. 


In 2004, the National Centre of Science, Information and Communication 
Technology, and Mathematics Education for Rural and Regional Australia 
(SiMERR National Centre) received an establishment grant from the Australian 
Federal Government. This remains one of the largest education grants awarded in 
Australia and indicates the importance attached to issues concerning rural and 
regional education. 

SiMERR was established at the University of New England (UNE) in Armidale, 
a rural centre, utilising a collaborative model involving groups of academics in 
each state (referred to as state Hubs). SiMERR carries out research and 
professional development activities with a focus on improving the learning 
outcomes of all Australian students, especially those studying in rural and regional 

Impulse for Initiative and Challenge 

The rationale for the SiMERR National Centre was based on compelling evidence 
from many sources (e.g., Programme for International Assessment (PISA), 
Thomson, Cresswell, & De Bortoli, 2004; Thomson & De Bortoli, 2007; the 
Trends in International Mathematics and Science Study (TIMSS), Zammit, 
Routitsky, & Greenwood, 2002; and national basic skills test information, 
MCEETYA, 2006) concerning the performance of students in rural and regional 
Australia, about a third of the Australian student population. 

These data quantified the extent of inequities for rural students in learning 
outcomes in science and mathematics education and underscore the most 
significant challenge currently facing education in Australia - equity of educational 
opportunity for all school students regardless of location (e.g., Lyons, Cooksey, 
Panizzon, Parnell, & Pegg, 2006; Roberts, 2005; Vinson, 2002). 

Table 1 illustrates one example of data currently available. Here the columns 
illustrate PISA summary data for Australia in 2003 and 2006 considered in terms 
of location. There are significant differences in achievement between students in 
each of these location groups. 



Table I . PISA 2003/2006 Mathematics achievement (mean scores) by location (Thomson, 
Cresswell, & De Bortoli, 2004; Thomson & De Bortoli, 2007) 

Average Score 



Australia Overall 



OECD countries 



Metropolitan Australia 



Provincial Australia 



Remote Australia 



Goals and Intervention Strategy 

SiMERR was established to carry out strategic and applied research, and work with 
rural communities to achieve improved educational outcomes for students. 
The vision of the work of SiMERR is formulated in three can-do-statements: 

- Parents can send their children to rural or regional schools knowing they will 
experience equal opportunities for a quality education; 

- Students can attend rural or regional schools realising their academic potential 
in Science, ICT and Mathematics; and 

- Teachers can work in rural or regional schools and be professionally connected 
and supported. 

To achieve this mission, SiMERR programmes identify and address important 
educational issues of (i) specific concern to education in rural Australia, and (ii) 
national concern in mathematics, science and ICT education across Australia by 
working in rural schools. 

Implementation and Communication 

SiMERR members are involved in approximately 120 projects. While some 
involve small numbers of schools (often in remote areas), teachers, and students, 
other projects span across regions or state jurisdictions. 

Many projects have national relevance, not only for rural areas but also more 
broadly for all Australian students. It has become clear that in working to address 
the needs of rural students, the findings and solutions that are emerging offer ways 
of enhancing student-learning outcomes in metropolitan areas as well. In an 
exemplary way, this is sketched below in brief descriptions of five large-scale 

/. National Survey of Issues in Teaching and Learning Science, ICT and 
Mathematics in Rural and Regional Australia (Lyons et al., 2006). 
This project involved extensive questionnaire surveys of teachers and parents of 
students from primary and secondary schools across Australia. Every provincial 
and remote school, and a sample of metropolitan schools, in Australia were invited 
to participate in the survey. Focus group interviews were conducted with a 



representative sample of teachers, parents and students from rural schools in each 

The survey data provided critical information about key themes that are 
considered to be limiting student outcomes in mathematics for rural and regional 
Australia as well as offering some practical ways of addressing these issues. The 
recommendations focus on several key areas including: 

• Staffing issues such as attraction and retention of teachers; 

• Teacher training and qualifications; 

• Professional development needs of teachers; 

• Resource material needs of teachers; 

• Learning opportunities and experiences of students. 

2. Identifying and Analysing Processes of Groups of Teachers Producing 
Outstanding Educational Outcomes in Mathematics (Pegg, Lynch, & Panizzon, 

This project explored factors leading to outstanding mathematics outcomes in 
junior secondary education for students across the ability spectrum. The focus was 
on the characteristics of and processes used by groups of teachers. Mathematics 
faculties achieving outstanding student-learning outcomes were identified by 
drawing upon extensive quantitative and qualitative data-bases. The study involved 
intensive case studies to identify faculty-level factors. Seven common themes are 
reported and these are the strong sense of team, staff qualifications and experience, 
teaching style, time on task, assessment practices, expectations of students, and 
teachers caring for students. 

The research highlighted a number of potential important issues for schooling 
into the future around the needs: 

• To provide opportunities to help teachers develop the knowledge and skills 
necessary to exercise effective leadership in the role of faculty leader; 

• For early career teachers to work with and learn from experienced mid and later 
career teachers; 

• To facilitate strong group interaction within faculties; 

• For relevant school-based professional development; 

• For high subject-knowledge standards for new and current teachers; 

• To create a culture in which teaching and learning, rather than behaviour 
management, dominates all classrooms; and 

• To develop common goals among teachers, students and the local community. 

3. QuickSmart intervention programme for middle-school students performing at 
or below National Numeracy Benchmarks (Pegg & Graham, 2005, 2007) 

This research programme is referred to by the generic title QuickSmart because it 
teaches students how to become quick (and accurate) in response speed and smart 
in strategy use. This teaching programme sought to improve automaticity, 
operational ised by students' fluency and facility with basic mathematics facts for 
those students in their middle years of schooling below national benchmarks. The 



programme refers to intensive focused instruction associated with the students 
being withdrawn in pairs from class for three periods a week over a 30-week time- 

The results found that improving automaticity in basic skills frees up working 
memory processing, enabling students to undertake more advanced tasks that were 
not specifically focused on during the intervention programme and these positive 
effects are still in play years after the intervention. 

4. Maths: Why Not? Unpacking reasons for students ' decisions concerning higher- 
level mathematics in the senior secondary years (McPhan, Morony, Pegg, 
Cooksey, & Lynch, 2008) 

The project considers why many capable students are not choosing to take higher- 
level mathematics in the senior years of schooling. This lack of numbers runs 
counter to the national need for a highly skilled workforce to remain competitive in 
the global knowledge economy. Australia is facing a multi-faceted skills shortage 
just when there is a need for more students to leave school with a sound grounding 
in higher mathematics. 

The results provide an important "toehold" to a number of critical issues 
underpinning the learning and teaching of senior mathematics in Australia. More 
importantly, it offers a means of connecting the learning and teaching of 
mathematics from the perspective of current and projected skills shortages. The 
project offers new insights into the problem and a platform for constructive 
national action. 

5. Collaborative innovations in rural and regional secondary schools: Enhancing 
student learning in mathematics and science (Panizzon & Pegg, 2008) 

This project created networks of rural teachers to form learning communities in 
science and mathematics. Each team of teachers in a particular school identified an 
important issue they believe was impeding student learning within their own 
school. This issue became the focus of the professional learning. 

Teachers were supported at an optimum time with help varying from school to 
school depending on the needs of the staff and students. Support was provided by 
(i) consultants with expertise in curriculum, assessment, and quality pedagogy 
visiting and working with the teachers at key points during the eighteen months of 
the project, and (ii) teams of teachers met on a few occasions to share their 
experiences with other teachers involved in the project. These meetings were 
crucial because they facilitated opportunities for teachers geographically isolated to 
meet collectively and communicate their ideas, challenges and successes. The 
model of professional learning used was seen to be highly relevant and cost- 
effective for schools that were widely separated by distance. 

Evaluation and Impact 

Evaluation of SiMERR occurs through two separated but related processes. The 
first concerns sets of agreed milestones concerning progress on a six-monthly 



basis. These targets were mutually agreed to, and offer a broad context within 
which SiMERR attempts to address its mission. The second process is related to 
individual projects undertaken by academics associated with SiMERR, including 
those internally financed through targeted funds within the Centre or from 
successful contracts with funding bodies outside of SiMERR. 

Tying down "impact" in such a diverse area is fraught with problems. At the 
heart of the work of SiMERR is building a network where teachers, educators, 
universities, education authorities, and communities can reflect and initiate actions 
on improving the current situation in rural areas for teachers and students. 

There are important signs that projects are having an influence. A critical 
purpose of these approaches is to have an evidential basis from which informed 
policy decisions can be made on how funding and actions might best target the real 
learning needs of different groups of students. In terms of the five projects outlined 
above there are now: 

• Recommendations to advise Federal policy as it relates to addressing inequity 
in rural students learning outcomes as a result of the SiMERR National 

• Published books identifying characteristics of faculty departments achieving 
outstanding educational student learning outcomes across the student ability 

• Recommendations to guide Federal Government policy on ways to encourage 
and facilitate more senior secondary students to undertake high-level 
mathematics courses; 

• Solid evidence that students (including Indigenous students) who have been 
performing at or below national benchmarks in numeracy for many years can 
be supported and show considerable improvement in basic mathematical 
skills and understanding; 

• Evidence of the nature of the successes for rural schools in solving issues 
relevant to them in teaching mathematics and how this professional learning 
can be encouraged and sustained. 

As a result of SiMERR activities there is now: a large number of research 
activities that have been awarded to academic groups (SiMERR Hubs) to support 
rural schools, teachers and students; a stronger national awareness and a higher 
media presence about rural concerns in education; and stronger support for 
professional teaching associations to provide more targeted professional support 
for teachers in rural locations. 

Challenges and Further Steps 

SiMERR has sought to influence positively the educational outcomes of rural 
students whose educational opportunities do not match those of their metropolitan 
counterparts, and to reduce the professional isolation of teachers. This has been 
pursued through targeted research programmes to inform education policy, 



teaching practice and pedagogy, professional development programmes, and 
teaching and learning interventions for teachers and students. 

Engendering and maintaining a climate of collaboration and trust among 
universities and their staff, education jurisdictions and their schools, teachers and 
communities around the country is critical to the success of the SiMERR operation. 
The capacity to engage schools to participate in activities is built on networking 
with teachers, education authorities and professional education organizations. 
These fruitful connections are important in building trust and rapport between 
schools and researchers, and they also facilitate discussion and collegiality. They 
are also critical players in attempts to move the findings of research to scale (see 
Cobb and Smith, this volume). 

The model of collaboration developed by SiMERR is in contrast to the highly 
competitive practices of universities in other fields of endeavour within Australia. 
It is recognised as important for the long-term that individual universities are 
supported to maintain and celebrate their own integrity, identity and successes as 
well as those achievements of the collective. 


South Korea has achieved extremely strong national growth that has resulted in 
rapid economic development since the 1950s. However, Kitawaga (2006, p. 15) 
pointed out that although dramatic, the post-war revival in South Korea was more 
about the developments in Seoul rather than more broadly across the nation. A 
recent consequence of this imbalance has seen the Government launch major 
decentralisation reforms with strong regional development policies. 

In order to address this issue in 2004, the South Korean government allocated 
over one billion US-dollars and embarked on a series of projects referred to as the 
New University for Rural Innovation (NURI) initiative. At the Kongju National 
University NURI funds were allocated to address rural education issues. In 
particular, the focus was on the development of a new programme as part of the 
Bachelor of Education programme for prospective secondary teachers. 

Impulse for the Initiative and Challenge 

In the PISA results of 2003, South Korean fifteen year olds were in the top group 
of countries in science, mathematics and problem solving, placing them second 
overall (OECD, 2006). However, the same PISA data show that there is a low level 
of satisfaction towards schooling, and parents have their children undertake 
extensive learning activities out of school time. It is estimated that 73% of students 
in primary and secondary education receive private tutoring after school hours with 
an additional 2.2% of GDP (Gross Domestic Product) allocated to private tutoring. 
Private outlays for education in South Korea are the highest of any OECD country 
(OECD, 2006, p. 29). 

Also of concern was the achievement gap between students enrolled in rural 
schools as compared to those in urban areas. While the average score in 



mathematics for South Korean students was 542 compared to the OECD average of 
500, those students who lived in communities of 3,000 or less had an average score 
of 447. This score compared unfavourably with the OECD average for small 
communities of 477. These data in mathematics were further confirmed (Im, 
2007a, p. 99) when an effect size gap between rural and regional students of 0.62 
was identified. 

Goals and Intervention Strategy 

The NURI project includes: 

- A programme for developing ICT pedagogical skills, involving effective ways 
of using computers in education as well as the establishment and management of 
teaching and learning systems for e- learning. 

- A programme for developing understanding of rural societies and rural schools 
by exploring issues of rural education more explicitly. 

- A programme to help prospective teachers adjust to rural schools and rural life 
by volunteering for service in rural communities. This involves voluntary work 
in educational contexts, practicum in rural schools, inviting rural school students 
to university campus, development of a practicum manual for classroom 
teaching in rural schools. 

- A programme for enhancing pedagogical skills in classroom teaching built 
around microteaching and the establishment of two laboratories for analysis of 
classroom behaviours. 

- A programme for learning foreign language, improving teaching school subjects 
in English, visiting rural schools in foreign countries. 

- A programme for enhancing teacher knowledge and enhancing success rates for 
securing teaching positions. 

Implementation and Communication 

This programme was developed as the education part of the four-year NURI 
project and was based on NURI-TEIC (Teacher Education Innovation Centre) 
team's meta-analysis (Im, Lee, & Kwon, 2007) of previous studies with Korean 
participants on the educational gap between rural and urban areas in Korea. The 
programme designed built on the current programme for prospective teachers with 
an additional emphasis on developing teachers' classroom skills, involving 
learning and practising ICT, and learning to understand rural societies and schools 
in a deeper way. 

Findings from the implementation program are discussed regularly with 
representatives of the seven Departments at Kongju University as well as 
colleagues at other universities who are interested in or who have made 
contributions to the initiative. Also the NURI-TEIC team maintains strong 
cooperative relationships with over 80 community schools, the Chungnam 
Provincial Office of Education, and the Kongju city office. 



Evaluation and Impact 

There are numerous ways in which the initiative is being evaluated (Im, 2007b). 
Most important is that the number of prospective teachers who are passing tertiary 
examinations is increasing over the rate in years prior to the advent of the 
programmes. There has also been a commensurate increase in the number of other 
certificates of attainment, such as ICT competencies, than in the past. 

As part of the evaluation of the impact of the NURI project, the prospective 
teachers and secondary students have undertaken surveys. In both cases, the results 
of the surveys have identified improvements in perceptions. Prospective teachers 
have responded positively to the changes in the course finding it more useful and 
relevant than in the past. Secondary students reported that the prospective teachers 
were more knowledgeable about them and their communities and they were more 
appreciative of the efforts of the prospective teachers than in the past. 

Challenges and Further Steps 

The NURI project at Kongju National University is set to finish during 2008. 
However, plans are in place to apply for a further 5-year (2009-2014) post-NURI 
grant to the South Korean government. The strength of this new application is on 
the track record evidence accumulated. Clearly, many of the benefits of the current 
funding in terms of course structures, resources and data on prospective teachers' 
development will still be available. However, it remains to be seen at this stage 
whether all the initiatives currently being undertaken can be maintained in the 
absence of such funding. 

One of the great challenges lies with the evaluation of the initiative. Firstly, it 
takes time for the full effect of a programme, such as described, to be felt. 
Secondly, outcomes in complex areas such as addressing issues associated with 
poor student learning in rural education are subject to many competing and 
complex interactions. These often fall outside of the education focus of the 
intervention and have much to do with the socio-economic viability of the 
particular rural area. Hence, attributing success of a program or otherwise is 
difficult. These issues increase the complexity of providing justifications for the 
spending allocated and proof that tax-payer money is not wasted. 


The motivation for the initiatives described in this chapter was a perceived 
deficiency in mathematics (and science) skills of particular groups of students, 
following large-scale international surveys or state-wide surveys revealing regional 

The surveys also revealed some structural problems with the overall education 
system. In Austria, it was mostly the system's fragmentary nature and lack of 
researchers in science education. In Ohio, they identified a lack of overt standards 
of education upon which to make judgements. In Australia, there was no specific 



national research body or a comprehensive national agenda dedicated to improve 
the learning outcomes of rural students. For South Korea, it was the difficulty of 
attracting and retaining teachers in rural areas. 

In all four cases, a government intervention followed. Improving the teaching 
and learning of mathematics and science became a matter of national policy and 
funds were allocated to begin to address the situation. National or regional centres 
were established and initiatives to improve standards or to address inequities 

Participants and Their Roles 

The relevant environments and participants in these initiatives were the federal and 
state government bodies (which financed and oversaw these initiatives), education 
experts from selected universities and (in the case of Australia) professional 
teaching organisations, and targeted schools (their teachers and students, partially 
also parents and community groups). Significantly, all initiatives build on forming 
regional and district structures incorporating local stakeholders (see also Cobb & 
Smith, this volume). 

In three countries, a particular university was given the role to set up a centre, 
which coordinated the entire initiative. In the Australian and the Austrian case, the 
leading university linked, through a tender process, to the involvement of academic 
staff from several other universities and other institutions throughout the country. 

Initial participation in all four countries involved only particular schools and 
particular segments of the student population. In Austria, initially it was upper 
secondary schools and later extended to all secondary schools, and finally also to 
primary schools, in Ohio students of middle-school years, in Australia samples of 
rural and regional schools and teachers were chosen for different initiatives, and in 
South Korea secondary schools were involved. 

Goals and Intervention Strategy 

In all four countries, the overall goal was to improve the teaching and learning in 
mathematics and science by improving the teachers' skills, establishing teacher 
networks through which teachers could communicate with each other and with 
education experts and, in some cases, establishing a nation-wide support system 
(Austria) or state-wide standards (Ohio). 

In Austria, the major goals were: the initiation, promotion, dissemination, 
networking and analysis of innovations in schools (and to some extent also in 
teacher education institutes) and recommendations for a support system for the 
quality development of mathematics, science and technology teaching. Innovation 
was the key word; participation was voluntary. Teachers and schools defined their 
own starting points and goals and were then supported by researchers and expert 
teachers. The emphasis was on supporting teams of teachers from one school rather 
than individual teachers. The teachers and schools retained ownership of their 
innovations. Another important aim of the Austrian initiative was networking. 



In Ohio, the overall goal was to improve the mathematical and scientific 
knowledge of students, and to achieve equity among students of different ethnic 
groups as well as gender equity through inquiry-based instruction. The Discovery 
project also aimed to develop state-wide standards for mathematics and science 
teaching. This involved developing shared methods of teaching science and 
mathematics rather than standardising a curriculum. A particular focus was on 
educating the teachers (through Summer Institutes) in how to implement inquiry- 
based instruction in the classroom. 

In Australia, the goal was to enhance teacher growth in rural and regional areas 
and to maximise high levels of teaching competence and student learning outcomes 
in the critical subject areas of mathematics, science and ICT. An additional goal 
was to set up teacher networks to help address professional isolation. The focus on 
science and ICT in addition to mathematics gave teachers a greater chance of 
interacting with a critical mass of teachers. 

The South Korean initiative aimed to enhance pedagogic skills of prospective 
teachers. Focus was primarily on improving their e-learning skills, their 
understanding of rural societies and rural schools, and enhancing their 
microteaching skills. 

The theoretical frameworks used differed among these countries. However, all 
these programmes built on the assumption that teachers play a key role in the 
intended change; thus they (as well as their students) were seen as active 
constructors of their knowledge. In Austria, the theoretical framework built on the 
ideas of action research and systemic approaches to educational change and system 
theory. In Ohio, it revolved around inquiry-based instruction. In Australia, it 
involved implementation of teaching methods and approaches supported by 
empirical evidence through ongoing research into teaching mathematics and 
helping move these activities to scale. In South Korea, various programmes to 
enhance teacher knowledge and microteaching skills, as well as their sociological 
understanding of rural schools and communities drawn from empirically based 
information from the research literature. 

The four countries used various intervention strategies to achieve their goals. 
The broader the goal the more strategies were used. Thus in Austria the primary 
strategy was to support teams of teachers from a particular school through different 
programmes. The broader focus was on promoting innovation, dissemination of 
knowledge, networking, carrying out analyses of innovations, and building a 
sustainable support system. In Ohio, the primary strategy was to involve teachers 
in Summer Institute programmes to improve their competence and content 
knowledge. An additional focus was on developing standards. In Australia, the 
intervention strategy involved collaboration with communities, educational 
authorities, professional associations and industry groups with the aim to develop 
solutions to problems faced by teachers, particularly those who are professionally 
isolated. In South Korea, the main strategy comprised programmes of pre-service 
education to complement and extend the traditional mathematics preparation 
programme and programmes to help prospective teachers adjust to rural schools 
and rural life. The latter involved practice-teaching periods in rural schools, 



inviting rural school students to university campus, and development of practice- 
teaching manuals for classroom teaching in rural schools. 

Collaboration, Communication, Partnership as Central Notions 

In all four countries, there was a strong emphasis on collaboration and 
communication between various social agents during all stages of the initiative. 
Long-term collaboration was seen as an essential basis for reform and 
sustainability of any reforms. By comparison, working on short-term professional 
learning activities was implicitly understood as being ineffective and highly 
unlikely to result in sustained improvements and teacher growth. 

Collaboration in Austria took the form of a wide network of people and 
institutions involved in the initiative. Staff and members of IMST supported large 
numbers of teachers over several years. To facilitate communication between 
teachers they also worked out inter-disciplinary connected concepts for basic 
education at the upper secondary level for four subjects. There was collaboration 
among teachers and staff members from the first meeting during which goals and 
research questions were established to guide later analyses of the effectiveness of 
new teaching methods. Teachers also helped each other as "critical friends". The 
various programmes operated as small professional communities that supported 
each participant. Furthermore, regional networks were established through which 
experienced teachers were able to disseminate their knowledge to other teachers. 
Communication was also facilitated by the IMST website which includes all 
documents written by staff members and teachers, by an annual conference, and by 
a quarterly newsletter. 

In Ohio, the collaboration brought together in-service teachers and university 
researchers and science educators who constructed professional development 
courses and taught in those courses. The teachers participating in the programme 
observed each other teaching classes and providing feedback to each other. After 
the course they met several times during the following year and exchanged their 
experiences. The emphasis on collaboration and communication also extended to 
classroom practice where inquiry-based methods encouraged interaction and 
communication between students. The collaboration also involved the partnerships 
between teachers and parents in science education. Finally, the Discovery project 
also led to the establishment of multiple professional development initiatives and 
networks of K-16 institutions through a variety of state and national funding 

In Australia, the emphasis on collaboration led to the formation of a National 
Centre at the University of New England and the nation-wide network of "hubs" 
located at several universities. The hub members communicated with each other 
and regularly met to share ideas. Each hub has its own website with hyperlinks to 
other websites and to SiMERR website. Hub coordinators, disciplinary groups and 
project teams also regularly conducted video meetings. In 2005 and 2007, the Hub 
members and relevant stakeholder organizations met at National Summits. Other 
meetings of members across Hubs coincided with disciplinary-based conferences 



or workshops. The collaboration and communication between various stakeholders 
was also stimulated by a higher media profile of issues of rural and regional 

Collaboration and communication is also an important issue in the South Korean 
initiative. It involves the relevant branches of the South Korean government, 
education experts at Kongju National University, educators with common interests 
from other universities, province education authorities, Kongju city office, and 
teachers. The purpose of the work with these groups is to inject different 
perspectives into the development and implementation of the program as well as 
ownership of ideas across a broad base helping ensure greater flexibility and 
cooperation at all levels of society. 

In all cases, the exchange of experiences played a predominant role among 
teachers, among university staff, as well as between teachers and university staff. 

Evaluation and Impact 

Evaluations were an integral part of each initiative and these evaluations revealed 
positive outcomes. In Austria, evaluation processes were a central part of IMST 
from its inception. The focus was on process-oriented, outcome-oriented, and 
knowledge-oriented evaluation. It involved both self-evaluation and evaluation by 
external experts. Independent international experts also assessed the overall 
effectiveness of IMST. The results of these evaluations indicated that the intended 
goals were achieved in the most part as well as possible improvements for the 
continuation of the project. Answers to teachers' questionnaires indicated a high 
level of satisfaction with the programme. Recently, more research on students' and 
teachers' beliefs and growth is being carried out. 

In Ohio, the evaluation of the programme centred on the test results of students 
in the Discovery Project as compared with students/classes who were not involved. 
Students in the programme performed better. Further, when school cohorts were 
compared, schools with a higher percentage of Discovery trained teachers also 
performed better than those schools that had fewer trained teachers in the 

In Australia, members of SiMERR have been involved in approximately 1 20 
projects throughout Australia across the four discipline areas. The results so far 
indicate that projects are beginning to have a major influence on teacher 
professional learning resulting in improved student-learning outcomes with 
increased effect-size measures. Interestingly, in a number of cases, data are 
showing that the policy announcements, programmes and the solutions developed 
for rural areas are also having an impact when implemented on student-learning 
outcomes in urban areas. 

The evaluations of the NURI project in South Korea show an increasing number 
of prospective teachers who are passing tertiary examinations. Surveys of both 
prospective teachers and students indicate high levels of satisfaction. Teachers find 
the courses relevant and secondary students report that the teachers are more 
knowledgeable of them and their communities. The midterm report of the NURI 



project received strong endorsement from the Ministry of Education of South 

All projects delivered positive results that demonstrated growth for participant 
teachers and their students. These data appear to have had a three-fold impact, on 
policy, institutions, and the way teachers fundamentally work. These examples 
make the point that when programmes are well developed and involve 
collaboration, communication and learning partnerships, real changes can be 
expected. However, crucially they still depend on continuing government support 
and in most cases the allocation of funds. 

The experiences documented above indicate that the presence of an intensive 
evaluation not only allows stakeholders to react to the results of the evaluation 
during a project (in the sense of a formative evaluation) but also - partly related 
with the former aspect - increases the likelihood that a programme (or parts of it) is 
prolonged or enlarged, or implemented in the system in one form or another. 


What are the lessons learned from comparing these four cases? Although, 
challenges arose from similar sources of data (e.g., studies like TIMSS, PISA, or 
national tests), each country has its own genuine context and specific strengths and 
weaknesses. Also, those people charged with leading changes in these countries 
bring different skills, and knowledge and belief sets to each enterprise. Thus 
projects invariably evolve to take different forms, emphases and directions. 
However, in all programmes, collaboration, communication, and partnerships 
were seen to have played a major role, not only between teachers and university 
staff members of the programme but also within these groups. 

Close collaboration was an important aspect among stakeholder groups formed 
by the ministry (policy, funding), the school practice (teachers), and the scientific 
community (teacher educators, researchers). An intensive kind of evaluation 
yielding relevant data to all parties concerned and the discussion about the results 
seems to enrich the quality of the project and contributes to its further support by 
helping to shape policy decisions. 

Communication occurred in: oral forms (e.g., workshops, seminars, conferences, 
and network meetings); written forms (e.g., newsletters, reflective papers by 
teachers, other publications); and electronic forms (e.g., materials on a website, 
chat-rooms, emails, etc). This open approach is supported by research on 
"successful" schools showing that such schools are more likely to have teachers 
who have continual substantive interactions (Little, 1982) or that inter-staff 
relations are seen as an important dimension of school quality (Pegg, Lynch, & 
Panizzon, 2007; Reynolds et al., 2002). Similar results can be found in other 
national programmes (e.g., in Germany, Prenzel, & Ostermeier, 2006). 

A very important issue concerned the insight of the significance of partnerships. 
Teachers were not only seen as "participants" of teacher education but as crucial 
"change agents" of the education system, regarded as collaborators and experts. 
Consequently, they were expected and encouraged to take an active role in their 



professional growth. Teachers were critical stakeholders who were themselves 
learners in the process of bringing about improved learning environments for 
students. Teachers needed to be sensitively supported by teacher educators through 
research-based advice and evaluation that gave meaningful feedback to teachers 
and generated new scientific knowledge. 

Specific recommendations are not easy to make since conditions and contexts in 
these countries are very different. However, it seems worthwhile that in any major 
intervention the following questions are considered: What kind of active role do 
national programmes ascribe to their teachers as change agents? How can exchange 
of experiences among teachers and researchers be promoted? How can 
communication (e.g., using stakeholder networks) and infrastructure (e.g., national 
centres) be established and further developed? How can evaluation contribute both 
to the improvement of the projects' process and their impacts as well as to the 
generation of scientific knowledge (which in turn contributes to the improvement 
of interventions)? How can the collaboration between different stakeholders of 
nations' educational change, above all, policy makers, teachers, and researchers be 
designed in a way that all parties - including the students and parents - feel 
empowered by national initiatives? How does the involvement in such (mostly) 
larger intervention projects change the role of researchers? What form should 
evidence take to provide reliable feedback about the success or otherwise of the 
initiative? How can the sustainability of innovations be supported and evaluated 
(see also Lerman & Zehetmeier, this volume)? We believe that mathematics 
educators need to be active participants in the discussion of these kinds of 


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John Pegg 

National Centre for Science, ICT and 

Mathematics Education for Rural and Regional Australia 

University of New England 




Konrad Krainer 

InstitutfUr Unterrichts- und Schulentwicklung 

University ofKlagenfurt 











In recent years, action research has found its way in teacher education and 
mathematics teacher development in particular. In this article, we introduce action 
research in its various conceptions. We go on to present studies on the way action 
research is used in prospective and practising teacher development. The article 
addresses issues educational researchers have to attend to, when supporting 
teachers in engaging in action research projects, and discusses what accounts of 
action research should attend to enable future meta-analysis on the impact of 
action research. The article includes two examples of a support system of action 
research, one from Austria, detailing an organizational structure to enable many 
teachers at different schools to engage in action research, one from the Czech 
Republic, presenting the results of a project of close collaboration of teachers with 
researchers, engaging in joint reflection. 


In the last two decades action research has seen a revival in the educational 
community at large (see e.g., Adler, Ball, Krainer, Fou-Lai, & Novotna, 2005; 
Kramer, 2006). 1 With new research on educational change small and large, the 
important role of the participants in enabling, shaping and maintaining change 
processes has become more and more recognized (Fullan, 2001; Wagner, 1997). 
For example, the Czech psychologist, Helus (2001, p. 37) emphasized: "A 
successful effort to change the school is only possible if the teacher becomes its 
leading agent". Seeing the practitioners at each system level as the pivotal figures 
of change processes, it is not surprising that action research is seen as one lever to 
better practices in education. Action research promises to support the change of the 
most important change agents, to ground changes locally where change is 
necessary, and to bring about personal growth that affords the retention of the 
pursued changes. 

In this chapter, we will explore the use of action research in professional 
development generally and then specifically in mathematics education. To that end, 
we will first introduce action research and then discuss the educational context of 

' For example, in their presentation of the discussions of a thematic group on teacher education of the 
European Research in Mathematics Education conference (ERME ), Krainer and Goffree conclude that 
they see an "|l]ncreased importance of action research as the systematic reflection of practitioners into 
their own practice" (Krainer & Goffree, 1999, p. 230). 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 283-307. 
© 2008 Sense Publishers. All rights reserved. 


action research: What are the problems action research is meant to solve, what is 
the context of action research? Next, we will elaborate on different conceptions and 
core elements of action research, to provide a framework for the subsequent 
discussion of action research in mathematics education and in the professional 
development of mathematics teachers. The examination will be completed by the 
presentation and discussion of two programmes or projects that use action research 
for (practising) teacher training in Austria and the Czech Republic. Finally, we will 
raise concerns and issues about using action research in teacher development in 
mathematics education. 


Action research is most frequently traced back to the work of the sociologist Kurt 
Lewin (1948), who constructed a theory of action research in the 1940s, although 
elements of the theory can be found earlier the writings of John Dewey (for a 
discussion of pre-cursors see Masters, 2000). Lewin incorporated key elements of 
today's action theory. He pointed out the importance of the participation of 
practitioners in all phases of the research process, and saw action research as a 
cyclical process of planning, action, and evaluation giving way to further planning, 
action and evaluation. Action research in education in particular is typically traced 
back to the teacher-researcher movement of Stenhouse (1975) who envisioned 
teachers taking an active role in curriculum development. Yet, action research - 
under the guidance of researchers - was already used in the field of curriculum 
studies in the late 1940s and early 1950s. Since then, action research in general 
(and in various forms, see below) has formed its own specialized community 
within educational research, with heterogeneous strands, since local (national) 
developments are strongly influenced by nationally prevalent theories (e.g., the 
approach of "Handlungsforschung" in Germany). Altrichter (1990) presents an 
account of the way action research fits into the framework of academic research in 
general. As of today, there is no unanimous definition of action research, Adler 
(1997, p. 99) even argues that there is a "war of definition" being waged. Given the 
different perspectives and associated claims and value judgments (see below), she 
and others (e.g., Jaworski, 1998) make the case that other terms like "practical 
inquiry" are more appropriate. 

However, action research conceptions share a number of characteristics, even 
though they may differ in the importance of the adherence to any of them. Other 
aspects are notable in that they are contested or changing. In the following, we will 
present such characteristics and aspects. 

(1) In any conception, action research is practical in the sense that it is a form of 
localized problem solving. Something in the local setting is meant to be 
understood and if needed, changed for the better. Closely tied with this general 
goal is the participation of the actors of the specific social system. This has a 
twofold background that may gain different emphases. On the one hand, there is 
the social theoretical stance, that participants have a unique access and 



knowledge about their local system, and are thus in a position to generate a 
more in depth-insight into their system (Cochran-Smith & Lytle, 1990, 1999). 
On the other hand, there is the theoretical and practical insight gained from 
organizational studies (Fullan, 2001; Weick, 1995), that successful change 
processes are dependent on the support of the agents of a social system. 
Engaging agents in action research not only generates different kinds of 
knowledge, but also transfers ownership of the resulting recommendations for 
change to these agents and increases the likelihood that positive change will 
happen. Thus, action research can be framed as libratory (Braz Dias, 1999; 
Gutierrez, 2002). 2 

(2) In practice, action research conceptions differ in how much responsibility is 
conferred to the practitioners, in our case the teachers. Different terminologies 
have been used to differentiate different conceptions of the interaction between 
for example, university based researchers and school based teachers. One 
extreme conception of action research portrays a more traditional relationship 
between researcher and practitioner in which the researchers pose a problem, 
"do the research" (in communication with the practitioner) and ask the 
practitioners to validate the results and implement suggested changes. Another 
form of relationship sees the researchers and the practitioners as true 
collaborators who are constantly in a process of a joint construction of meaning 
about the situation at hand, its problems and possible solutions. This position 
entails that the localized working theories of action of the practitioners are just 
as valued as established academic theories. Both practitioner and researcher are 
seen as having different ways of seeing the world, with the practitioners' 
perspective being validated by their histories of action. The quality of proposals 
for change are validated by the emergent practice (or falsified by the failure of 
the system to respond as expected). Finally, action research can also be 
understood as practitioner research without the necessary and continuous 
involvement of any academic researcher. Instead, researchers may be consulted 
at all phases of the action research process, but the ownership of the overall 
enterprise rests firmly with the practitioners. Of course, one may envision many 
forms of cooperation and collaboration in between these extremes. 

Accounts of actual action research usually show the teacher to be more 
active than in the first conception; teachers are at least thinking or reflecting on 
their own practices. But theoretical discussions on the breath of action research 
may frame almost any valued aspect of teacher participation in a research 
project under the heading of action research. In all these cases, the practitioners 
are called on to help to approach local problems with locally suited solutions. 

(3) A further important element of any conception of action research is the 
notion of reflection (Fendler, 2003), which can also be traced back to Dewey. 

For an early discussion of action research striving for a liberating education see Carr and Kemmis 



Elliott's definition of action research frames it as teachers' systematic reflection 
of professional situations aiming on their further development (Elliott, 1981, p. 
1). Despite the importance of this concept, there is little theoretical and 
philosophical exploration to be found about what exactly "reflection" is 
supposed to mean. Instead "reflection" is generally taken to be a self-evident 
expression, which stands in a critical juxtaposition to action. Theoretical 
explorations frequently take up Schon's ( 1 983) notion of reflection in action, 
reflection on action and reflection for action as well as reflection on reflection 
in action. With these notions Schon explores the relationship between 
"knowing" and "doing", touching on many of the issues, which were later 
discussed by Suchman (1987) and others exploring situated cognition. We 
always know more about a situation than we can express. At the same time, we 
never know everything. When a situation surprises us, and we are able to attune 
our plans or unfolding practice to address surprising or new insights into a 
situation, Schon is talking about "reflection in action". Reflection on action and 
reflection for action both take place outside the pressures of having to act at the 
(extended) moment of reflection. The first (reflection on action) is looking 
backwards, thinking about what happened. Reflection for action is looking 
forward, carefully considering plans for action (instead of just reacting). Finally, 
reflection on reflection in action captures the careful consideration of what 
happened in reflection-in-action. Why did something surprise us, what elements 
of our (automatic) practice made us not expect something, how did we deal with 
it, what made it (un)successful? An example: If one is planning a unit taking 
into consideration what may happen, one is reflecting for action. If the unit took 
place, and one ponders about reasons for unexpected student answers, one is 
reflecting on action. If a usually engaged student becomes disruptive in class, 
and one manages to think about why and take immediate action based on the 
assumed cause that she was frustrated due to a misunderstanding (rather than 
responding with some routine reaction to disruptive students), one is reflecting 
in action. If one later on reflects why one would come to the conclusion that she 
was frustrated, and whether the chosen strategy was effective, one is reflecting 
on reflection in action. 

One further important element of Schon's account of a reflective 
practitioner was his conception of a reflective practice as a practice in which a 
practitioner is engaged in a constant conversation with his or her problem 
situation. Reflection on action and reflection for action may lead to promising 
plans for action. But in practice, life will always surprise us; thus expert 
performance requires being able to adjust, to be flexible, to reflect in action - 
and to learn from our adjustments to situations by reflecting on reflections in 
action (Doerr & Tinto, 2000). The complexity of practice portrayed by Schon is 
a beautiful rendering of the situation teachers face in their daily lives, in which 
complex social systems with multiple actors make the outcome (or the process) 
of most plans quite unpredictable. 



(4) The content, action research is concerned with, varies greatly. In terms of 
subject area, action research projects were originally more prevalent in the 
humanities (especially concerned with literacy), but now many action research 
projects are also found in mathematics and the sciences. Nevertheless, if one 
looks into articles published in the journal Educational Action Research, one 
will frequently find no indication of the subject considered in an action research 
project in the abstract, and sometimes not in the entire article. 3 We believe that 
the omission of the subject area is not accidental. Action research as classroom 
teacher research will usually problematise some aspects of classroom practice. 
This puts the interaction between student(s) and teacher or just between students 
into focus. Thus, action research projects frequently set out with pedagogical 
issues of classroom management and organization, or the projects trial some 
previously conceived classroom innovation and seek to confirm or disabuse the 
beliefs inherent in the conception (e.g., Watling, Catton, Hignett, & Moore, 
2000). Ball (2000) points out that it takes time, experience, and self-confidence 
to see classroom problems as possible problems of subject didactics and/or 
ultimately as problems of a lack of subject knowledge. 

(5) The notion of reflection and action is also closely tied in with the conception 
of the place of beliefs in a theory of change. In general, action research projects 
follow explicitly or implicitly the assumption that - when considering the 
relationship between beliefs and practice - practice takes precedence. In other 
words, it is assumed that a change in beliefs will not necessarily bring about a 
change in (teaching) practices, but changes or difficulties in teaching practices 
may change beliefs about a situation (or lead to a quest for understanding and a 
new formation of beliefs) (Fullan, 2001). Thus, action research projects do in 
general not set out to teach someone available expert knowledge on any 
particular problematic issue. If teachers are collaborating with or guided by 
researchers, expert knowledge might take the form of already synthesized and 
contextual ly sensitive contributions or advice, the breath of possibly available 
information is usually not made explicit; and reports of action research projects 
frequently do not present a survey of available literature on the issue the action 
research project was concerned with. Knowledge imported concerns foremost 
methodological issues about the action research itself. How does one plan an 
action research project, how does one collect evidence and arrive at an 
interpretation? The theoretical section of action research projects frequently 
addresses this theoretical and methodological frame of "an" action research 
project in terms of its process characteristics. 

(6) Conceptions of action research differ also in the significance they assign to 
peer support and collaboration. Early reports of action research and reports of 
action research in mathematics teacher education in the Journal of Mathematics 

We have made a data-base keyword search for mathematics in the last ten volumes of EAR; the results 
show only 6 articles from 1997-2006. 



Teacher Education (e.g., Halai, 1998) present action research projects of 
individual teachers. In recent years, more and more reports are about 
collaborative action research, like action research projects of groups of teachers, 
and some conceptions consider such a cooperation as a necessary element of an 
action research project. No person lives for him- or herself, no practice can be 
transformed in a social vacuum. Sustainable change needs a community. While 
those conceptions do not refute outright that action research could be done by 
individuals, the emphasis on (peer) feedback and discussion with peers as a 
structural prerequisite for an honest self-appraisal, for example, for self- 
reflection, render solitary projects as less potential and hence less desirable. 

(7) A notion not frequently used or reflected upon in articles on action research 
in mathematics education, but which we deem as an important conceptual 
contribution is Elliott's notion of first and second order action research. With 
this distinction, Elliott (1991) makes the point that doing action research is 
different than facilitating practitioners in their efforts. Thus, researchers 
collaborating with teachers on action research project have a different job, and 
need to reflect on different issues than the teachers engaged in action research 
(Losito, Pozzo, & Somekh, 1998). 

In the last decades, we have seen the call for and implementation of "reforms of 
education" in many educational systems around the world. Fullan (2001, p. 37) 
argued that many of these reforms failed and that "change will always fail until we 
find some way of developing infrastructures and processes that engage teachers in 
developing new understandings". Furthermore, "[cjhange as a change in practice 
entails changing (1) materials, (2) teaching approaches, (3) beliefs" (Fullan, 2001, 
p. 70). "To change beliefs, people need at least some experience of new 
behavio[u]ral practices they can discuss and reflect on" (Fullan, 2001, p. 45). 
Action research offers a way to address this need. What is it about action research, 
which lets some researchers see so much promise to change education practice, 
while others set their stakes somewhere else? 

A strength of action research is that the "action" it is concerned with is the 
(behavioural) situation and the behaviour of the teacher engaged in improving the 
situation at the same time. Action research provides a voice for teachers to share an 
emic view of their experiences with other teachers and the research community at 
large. The sharing of experiences has raised a number of methodological questions. 
What is the nature of the story being shared, what makes a story worthwhile being 
shared? "When does a self-study become research?" (Bullough & Pinnegar, 2001) 
Is this research at all? And if it is to count as research proper, what does this entail 
(Altrichter, 1990; Melrose, 2001)? 

In general, from the point of view of some researchers, questions raised about 
self-studies (Feldman, 2003) - and thus about action research, which can be seen as 



a specific form of self-study 4 - are the same as those being raised about case 
studies, with the key question of "What is this a case for?", addressing 
general izabi I ity. A related issue is the ability to particularize the general (case) to a 
particular context or particular issues. 5 Where one community of researchers may 
ask the question: "What is this a case for?", another may ask: "Does this research 
present enough details about a situation, to tell us something worthwhile about the 
context we are concerned about (e.g., mathematics teaching)?" 

All of this, of course, presupposes the stance of the educational researcher or the 
research community asking for contributions of teacher research to comply with 
the standards of research in general, and thus it asks for a high level of (research) 
proficiency of the members of this discourse community. If teachers enter into the 
education research community, if they raise their own voice within this 
community, they have to adhere. Considering the place of teacher in the 
educational system, this asks for a lot of competence of any one individual. 

Educators who do not want to charge teachers with following all the 
prescriptions for doing "good, valid research" will frequently argue for substituting 
"action research" with less loaded expressions like "teacher inquiry" (Feldman & 
Minstrel I, 2000). On the one hand, this will lighten expectation about "action 
research projects": They can be conceived as "good worthwhile projects" without 
falling short of standards of research at universities. Teachers engaged in such 
projects do not need to feel lacking. On the other hand, there are reports of many 
other teachers who feel self-empowered by doing research themselves on par with 
academic educational research. Educators who do not see action research projects 
necessarily as doing "research proper" usually value action research projects for 
their local problem solving and the professional learning happening within the 
course of such action research projects. Thus, they see action research more as a 
means to professional development than as a means to produce general and 
generalizable knowledge into teaching and learning. 

Cochran-Smith and Lytle (1990) hold that teacher research is a different game, 
which follows different rules and standards of quality; it is a genre all by itself and 
should not be judged by time-honoured standards of educational research. 

A different approach, which we turn to now, does not see teacher research as 
(foremost) providing new insights into teaching and learning - even though those 
contributions are valued - but regards teacher inquiry as a way to foster 
professional development. In this perspective, teacher research is not an end in 
itself, but the means to accomplish something else, and debates on the validity and 
generalizability of teacher research miss the point. The question is not about the 
quality of the products of teacher research, but the changes doing teacher research 
brings about in practitioners and their practices. 

4 Strictly speaking, this is not entirely true but depends on the conception of action research as discussed 
above. If teachers are investigating their own practice, they are engaged in self-study. This does not 
imply that they want to change their practice. But if self-study is instrumental in a developmental 
process, it turns into action research. 

5 We are grateful to Konrad Kramer for bringing up this point. 



On a theoretical level, some conceptions of professional development (Bruner, 
1996; Climent & Carrillo, 2001; Helus, 2001; Jaworski, 2003; Krainer, 1996) 
integrate the important concept of reflection into the theoretical conception of what 
is required of a competent teacher - for example, they include the "competence of 
reflection" as a further core dimension of teacher competence theories (e.g., of 
Bromme, 1994; Harel & Kien, 2004). 

Action Research as a Means for Professional Development in Mathematics 

Judged from the number of articles on research methods in action research and 
teacher inquiry which appeared in the Educational Researcher, action research and 
teacher inquiry has encountered a growing interest in the last. As for mathematics, 
the Journal of Mathematics Teacher Education devoted an entire issue to action 
research and teacher inquiry in 2006 (JMTE 9.3); and action research and teacher 
inquiry features prominently in the European Society for Research in Mathematics 
Education (ERME), for example, in a special volume of the first CERME- 
proceedings (see e.g., Krainer & Goffree, 1 999) and the continuous special interest 
group on teacher education at its conferences. Nevertheless, most papers on action 
research that encourage professional development of practising teachers are not 
presented as such. Instead, the papers focus on the problems and gains of the action 
research project itself, with changing practices, beliefs and understanding of the 
teachers being only part of the parcel. On further reflection, this mode of 
presentation is not surprising: Presenting action research projects as professional 
development enterprises objectifies teachers, and portrays them as in need of 
change. Putting the teacher as the learner into the centre, creating a story of 
learning might promote a story of a previously lacking individual's development. 
Given the entire philosophy of action research, such a move would counter the 
very approach, which rests on an appreciation of the knowledge in practice, and the 
practitioner as a professional. Nevertheless, papers on action research do generally 
report on changes of belief and practice, even though reports may vary on how 
elaborate those accounts are. While we have not found a systematic survey on the 
outcome of action research for professional development, a recent series of surveys 
on continuing professional development published by the EPPI-Centre (Evidence 
for Policy and Practice Information and Co-ordinating Centre) in London sheds 
light onto possible outcomes of action research. 

Action research is prominent in studies on collaborative Continuing 
Professional Development (CPD). In their review of the literature on the impact of 
collaborative continuing professional development for teachers K-9, the CPD 
Review Group (Cordingley, Bell, Rundell, & Evans, 2003, p. 32) 6 reported that 
26% of all the surveyed studies employed action research. In this report, action 
research as a method to foster collaborative continuing professional development, 

6 The review is not particular to mathematics, but twelve of the 30 studies with a curriculum focus on 
mathematics; of the 1 7 studies selected for an in-depth review, six concern mathematics. 



is only second to Peer Coaching (30.5%) closely followed by Workshops (25%) 
and Coaching (25%). 7 The report (Cordingley et a!., 2003, p. 4) finds that "the 
collaborative CPD was linked with improvements in both teaching and learning; 
many of these improvements were substantial". The authors report benefits to 
teachers with respect to self-confidence as teachers, a heightened belief in their 
self-efficacy as teachers, an increase in the motivation for collaborative work, and 
an increase in the willingness to change their practice. Likewise, students 
demonstrated higher motivation, better performance, and more positive attitude to 
specific subjects as well as more active participation. The authors also point out 
some important features that seem to have been conducive to the attainment of the 
positive results (and studies lacking elements were less effective): 

> the use of external expertise linked to school-based activity; 

> observation [e.g., teachers visiting and observing each others classroom, 
or researchers videotaping a lesson as a basis for further - joint - 

> feedback (usually based on observation); 

> an emphasis on peer support rather than leadership by supervisors; 

> scope for teacher participants to identify their own CPD focus; 

> processes to encourage, extend and structure professional dialogue; 

> processes for sustaining the CPD over time to enable teachers to embed 
the practices in their own classroom setting" 

(Cordingley et al., 2003, p. 5) 

The report also highlights the importance of expert input, including subject 
input, if "an intervention [is] intended to achieve subject specific changes" 
(Cordingley et al., 2003, p. 6). Furthermore, it stresses the importance that teachers 
are in a position to work on their own expressed learning needs, which also entails 
the adoption of differentiation strategies (such that each teacher can truly work on 
his or her individual concerns), that the collaboration is sustained and that teachers 
find a place where it is save to admit needs (and report problems and possible 

While this review is very positive on collaborative continuing professional 
development arriving at recommendations which square with principles of action 
research, another review (Gough, Kiwan, Sutcliffe, Simpson, & Houghton, 2003) 
finds that "while student attainment and learning styles profit from reflection, self- 
directed learning, planned action and similar approaches, there is no clear evidence 
on whether these approaches influence or change the learner's identity, reflective 
capacity or their attitudes about learning" (Gough et al., 2003, p. 64). Yet, (changes 
in) identity and reflective capacity are constructs which are difficult to capture; 
generally reflection is still a much valued element of action research and teacher 
inquiry projects, more recent case studies on the use of reflection in professional 
development programmes are positive (Even, 2005; Scherer & Steinbring, 2006; 
Ticha & HoSpesova, 2006). 

' Note that any study may make use of more than one type of invention. 



In his review of models of professional development, Foreman-Peck (2005, p. 9) 
states: "Practitioner inquiry and research is a strong element in professional 
development courses for teachers, [and] an important part of teachers' personal 
professional development". The above mentioned report (Gough et al., 2003) also 
finds that 56% of the surveyed studies which featured professional development 
planning (PDP) reported on activities which occurred as part of some coursework 
(e.g., an action research project done as a requirement in a teacher education 
programme). The authors raise the question "whether the emphasis on course- 
specific outcomes in any way restricts the reflection that takes place as part of such 
interventions" (Gough et al., 2003, p. 45). 

In the following, we want to discuss and problematize different aspects and 
dimensions of action research that feature in the literature on mathematics teacher 
education. 8 

(1) Choice of a topic and temporal dimension. The choice of the topic of action 
research is closely tied in with the question of where the incentive is coming 
from to do action research. If action research starts with the teacher, issues 
focus on localized, specific questions. Sometimes (e.g., Watson & De Geest, 
2005) papers report that researchers defining an agenda were looking for 
volunteers. In general, little is said about processes and negotiations leading up 
to the collaboration. The same is true for the last case, when action research is 
done as part of some course requirement. From our own experience, we know 
that defining the problem such that it is "workable", that it becomes clear what 
issues one needs to consider, what evidence one should attend to and collect 
can be a difficult and time-consuming process. Additionally, questions and the 
corresponding action research programme may change in the process. At the 
same time, we found few reports on aspects addressing these issues. One 
notable exception is Feldman and Minstrell (2000, p. 448) who state that this 
first phase may take up to one year. Ball (2000) also observes that simply 
finding a topic may take a long time. These are important elements. We need 
to know more about reasonable time-spans, the time more and less 
experienced teachers need to become comfortable with action research or self- 
directed inquiry. 

(2) Authorship and theorizing the area of investigation. As discussed above, 
theories on action research stress the unique characteristic of knowledge 

8 For the review, we systematically screened all volumes of the Journal of Mathematics Teacher 
Education (JMTE); we looked at all articles for the last 10 years of Educational Action Research 
(EAR), which were returned by a data-base search for "mathematics" (6 articles, 2 book reviews); we 
included the major articles which were returned by a keyword search for mathematics and "action 
research" in the JSTOR-database (in June 2007); we looked at articles keyworded for 'action research' 
in the Proceedings of the Psychology of Mathematics Education (PME), 2007 (none!) and the 
appropriate sections of CERME). This was complemented by various - but not systematically collected 
- articles from the handbooks of teacher education, discussions on various aspects of teacher inquiry in 
the Educational Researcher and additional information, which is grounded in our local research 



generated through action research and that doing this type of research requires 
a different approach, which gives precedence to insights gained by a grounded 
or "bottom-up" perspective of a situation, clearly rejecting preconceived 
notions as might be engrained in theories brought in from the outside. Yet, as 
educational researchers, we still hope to "bridge the gap" (see also Jaworski, 
2006; Scherer & Steinbring, 2006), and that those implicit theories which 
guide the perception and interpretation of the action researchers become 
explicit and thus enahJB, on the one hand, a communication between insights 
generated in the situation, and understandings found through traditional 
avenues and, on the other hand, a deeper understanding of possible points of 
view of engaged and reflective practitioners. 

A theoretical-terminological framework strives to capture the different 
modes of cooperation between teachers and researchers, and thus which logic 
and aims was given precedence (the logic of a certain practice in a field, or the 
logic of research). Thus one may talk about participatory research (with 
teachers/practitioners taking part in a research project), about cooperative 
research (with teachers/practitioners cooperating, such that each may reach 
their individual goals) and collaborative research (in which both try to learn 
from each other and both strive to achieve their shared, negotiated goals). All 
conceptions have clear consequences for authorship, decision processes and 
the status of academic and practitioner theories. Theorizing what research does 
to a field this way, also highlights how issues of authorship, aims of the 
enterprise and considered knowledge are interwoven. 

Most papers written in peer reviewed journals end up being written by the 
researchers collaborating with teachers. 9 Thus, even if action research 
(depending on the definition) may strive for an emic perspective, reports on 
action research are usually presented "from the outside". This does not imply a 
second-order action research perspective, since only a minority of articles 
focuses on the particular practice of the researcher in enabling and supporting 
action research projects. Rather, most projects are still reported from an 
"objective", almost outside point of view. There are some exceptions (see e.g., 
the report of a teachers' book project in Ponte, Serrazina, Sousa, & Fonseca, 
2003), but generally we found that those people, who report about their own 
development, came into the process as a teacher being engaged with their own 
practice, and ended up becoming at least part-time educational researchers (as 
e.g., in the case of Deborah Ball, John Mason, and Jim Minstrell). Minstrel I (in 
Feldman & Minstrell, 2000) states in a by-line, that he became so interested 
that he earned a Ph.D. What does this imply about action research as a 
professional development incentive? And what (kind of) people will be 
attracted to action research? With respect to those questions, teacher inquiry 
required by some programme provides an interesting context, since this is one 

9 See also Adler, Ball, Krainer, Lin, F.-L., & Novotna (2005, p. 371) who found, that "Most teacher 
education research is conducted by teacher educators studying the teachers with whom they are 



avenue in which teachers may not have chosen to pursue by themselves. This 
is one group which did not necessarily self-select for the substantial 
involvement required when questioning and researching ones own practice. 
Yet, so far we lack reports which attend to the more reluctant participants, 10 as 
well as reports which describe failures in going through an entire cycle of 
doing action research (a notable exception is Nickerson & Moriarty, 2005). 
How does one have to design a setting which encourages people not already 
perceiving a need to take up action research, and supports them while they 
struggle through the processes? What makes them drop out? (Christenson et 
al., 2002) When does someone become disenfranchised? 

The theoretical grounding is quite heterogeneous. However, apart from a 
general grounding in core concepts of teacher education - like Shulman's 
(1986, 1987) account of teacher competences - there are two concepts which 
seem prominent: SchSn's (1983) account of reflection, and Wenger's (1998) 
notion of communities of practice. As noted by (Cordingley, Bell, Thomason, 
& Firth, 2005), most action research projects are done in a collaborative 
context. If the action research itself is not performed by a community of 
inquiry (Garcia, Sanchez, Escudero, & Llinares, 2006; Jaworski, 2006), it 
studies the impact of the action researcher on a community (of learners). In 
both cases, peripheral members of some learning community are learning to 
become full members of a (reflective, self-directed) community of 
practitioners. If- in the education context - the communities of practice (Lave 
& Wenger, 1991) is conceived as a community of inquiry (Jaworski, 2006), 
one can see action research becoming subsumed in a more encompassing 
concept which integrates the stances of action research with a social learning 
and social community point of view. Elliott (1991) has maintained that action 
research is a group effort. However, given the above mentioned "war of 
definition", with the resulting uncertainty what someone means when they 
speak of action research, as well as the critique ventured again the notions of 
"action" and "research" in "action research", we may see a (terminological) 
move to "communities of inquiry" which at the same time stresses the 
importance of having communities support change, and avoids the problems of 
using a contested notion. However, it remains to be seen how such a 
conception will deal (or exclude) the individual teacher doing research (or 
inquiry) on their own classroom without being embedded in a community. 
(3) Mathematical content. In the articles we found reporting on action research in 
mathematics teacher education, mathematics and mathematical concepts form 
an undercurrent, a background, which is frequently not elaborated on. 
Mathematical concepts turn up in discussed classroom transcripts, to explore 
teachers' thinking and practice (and to ground teachers' joint reflection, see 
e.g., Goodell, 2006) and to discuss missed and taken up opportunities to learn. 
Nickerson and Moriarty (2005) discuss the importance of subject matter 

10 Ross and colleagues discuss that low self-efficacy of teachers leads them to avoid engaging in action 
researcher (Ross, Rolheiser, & Hogaboam-Gray, 1999). 



knowledge for the practice of their teachers. In their report, they talk about 
Habor View teachers, who had a comparably high level of subject knowledge 
at the beginning of the project, and the Palm teachers, who did not (Nickerson 
& Moriarty, 2005, p. 133): 

Our analysis suggests that teachers' knowledge of mathematics affected 
their collective control over decisions related to the mathematics program. 
Habor View teachers felt empowered to alter the curriculum. Palm 
teachers did not feel that they could. Increased mathematical knowledge 
supported teachers' recognition of the need for assistance. 

While this claim seems plausible, one has to consider that teacher training 
differs vastly across the world. What are the necessary thresholds to empower 
teachers to feel comfortable to make the "right choices"? How much 
knowledge is "enough"? And how can we support teachers to become aware 
that they (may) have "enough" (in their context!) - despite recognizing that 
there is still more to be learnt? 
(4) Success and failure stories. In mathematics teacher education, action research 
projects are usually presented by a number of case studies which demonstrate 
successes (with examples focusing on subject didactical elements). Studies 
focus on "what happens" during the action research projects, and the resulting 
changes. To enable future systematic meta-analyses, reports should be more 
detailed about contextual elements like processes leading up to the action 
research projects (subject selection, negotiation of topics, relationship between 
researcher-teacher, kind and extent of external support) as well as elements of 
the unfolding processes (time-spans). So far, we know little about "drop-outs", 
possible particular characteristics about or histories of those teachers who 
volunteer to undertake action research projects, and of what happens "after". 
Not knowing about the necessary time investment for different groups of 
teachers (stratified by experience, attitude and other possible factors) as well 
as the related outcome, we cannot gauge the relative impact of different 
conceptions of action research or other similar approaches (e.g., self-study) as 
professional development. In particular, it is still an open question as to how 
much (little) input and direction one needs to provide, in order to reach the 
reported positive benefits. 


The following two examples present first a case from Austria of a support system 
for teachers to engage in action research projects of their own choosing. The 
second case is taken from the Czech Republic, and presents a project, in which 
teachers were supported to collaboratively reflect and further develop their 
mathematics teaching. Thus, both cases can be seen as being at very different ends 
of a continuum (or continuous space): in the Austrian case, teachers (individuals or 
teams of teachers) choose topics of their own interest to work on. Their reports are 



rarely concerned with questions of didactics of mathematics; much more common 
are action research projects which trial the use of some teaching method (e.g., 
group work, or self-directed learning environments). In the Czech case, 
practitioners were invited to work on several topics (application, grasping of 
situation, problem posing, and geometry) and teachers then decided to focus on 
part-whole relation and fractions. Thus, it was much more oriented towards a 
common topic, while supporting teachers in their work and individual 
development. And it was naturally more concerned with didactics of mathematics. 

Austria: The I MST Project" 

In order to promote teacher development in mathematics and the natural sciences, 
the Austrian ministry of education launched the IMST3 project (Innovations in 
Mathematics, Science and Technology Teaching, 2004-2009; Krainer, 2007) which 
provides as one measure a fund for teacher research. This fund succeeded the 
IMST 2 project (2000-2004), which already supported teachers doing action 
research (Altricher, Posch, & Somekh, 1993) of their classrooms, schools or 
educational aspects concerning an entire region (Krainer et al., 2002). In both 
phases, the IMST project invited teachers, and teams of teachers (of the same or 
different schools) to submit project proposals for a one year project (with the 
option to submit a proposal to continue or extend already running projects). With 
the IMST3 project (and a growth in size of participants), the application to the fund 
makes use of an already highly structured online submission form, which asks, for 
example, how gender issues are attended to, and how people intend to evaluate the 
results of their changes or the state of affairs. Project proposal workshops across 
the country offer advice for teachers who find the required definition of their 
projects difficult. Moreover, since 2006 it is possible to participate as a teacher or 
team with the purpose of developing a project (proposal) to be submitted the 
following year. 

Taking in account the primary area of expertise of the involved teacher 
educators, at the beginning the fund set out to support teachers of college-bound 
high school students and to a lesser degree 9th to 13th grade students in general. In 
2004, the call was extended to lower secondary schools. In 2007 elementary school 
teachers were included for the first time. Thus, in 2007, the call for projects 
became open to mathematics and science teachers at all grade levels. 

The project proposals are evaluated by educators and mentor teachers; projects 
may be accepted with recommendations for further explication or changes. Each 
year about 150 projects are accepted. The contract with the teachers provides them 
with a budget for project expenses (as defined in the project proposal) and a small 
monetary compensation. It requires them to participate in two workshops and to 
submit a project report at the end of the year. Teachers are invited to a start-up-day, 
in which they are introduced to their advising teams, a specific advisor teacher (one 

" Note that the article of Lerman and Zehetmeier (this volume) also reports on the IMST project. 
However, their presentation does not address the details of the fund presented here. 



for about seven projects) and two other team members. These advisor teachers are 
experienced teachers at schools or teacher education institutes. Within this year, 
they attend two project workshops of their choosing out of, for example, an 
orientation workshop, a writing-workshop, an evaluation workshop, or a gender 
workshop. Throughout the year, they are asked to hand in an "action plan" 
(including their evaluation), which is discussed with their specific advising teacher. 
Likewise, the project report at the end of the year may be commented on and be 
revised before it is accepted by the advising teachers, and eventually be published 
in the internet. Thus, the fond provides some direction while clearly setting out the 
requirement to have teams of teachers work on issues of their own choosing. The 
support of the advising teacher focuses mostly on running the project: stating clear 
goals, working out a plan to proceed, planning the evaluation, and writing up the 
experience. Above and beyond teachers can consult experts to work on additional 

In the four years of running the fund, we encountered many of the same issues 
and concerns noted in the literature above: 

• Action research requires a lot of time and energy from teachers. Thus, 
asking teachers to volunteer undertaking an action research project usually 
leads to a self-selection of already engaged and enthusiastic teachers. 

• In schools in which principals supported teachers and they were 
embedded in a community of like-minded teachers, teachers were able to 
effect substantial changes, for example, introducing a new student-centred 
feedback culture in the school, or introducing observations of each others 
teaching. In other schools, changes were restricted to the respective 

• In a series of interviews (Benke, Erlacher, & Zehetmeier, 2006), we found 
that experienced teachers, who had tended to self-critical ly question their 
own teaching, reported a heightened sense of self-confidence due to the 
project. They felt more assured that they were on the right track, which 
also allowed them to be more assertive in discussions with colleagues. 

• We did not observe Fullan's or Elmore's problem, but it is certainly 
something to keep in mind: "It is a mistake for principals to go only with 
like-minded innovators. As Elmore (1995) puts it: '[Sjmall groups of self- 
selected reformers apparently seldom influence their peers.' (p. 20). They 
just create an even greater gap between themselves and others that 
eventually becomes impossible to bridge" (Fullan, 2001, pp. 99-100 and 

• As has been found elsewhere, the projects dealt more with pedagogical 
questions than with content-related ones (as e.g., mathematics). In one 
case, in which a mathematics teacher educator invited teachers to use 
materials he had developed explicating core conceptual elements, many 
teachers stopped posing their own questions, they "executed" the 
materials without realizing that the materials did not require "standard" 
pedagogical or didactical approaches. In other words, starting with 
"teacher problems" affords side-stepping mathematical didactical issues. 



• In the lMST-fund projects, many teachers started out with a vision what 
they wanted to work on. Yet, in order to afford a systematic, data-driven 
exploration for the required evaluation, most of them still needed to 
further explicate "the problem" to make it concrete enough to be able to 
look for evidence for some claim. Other teachers - who had less 
experience with a research point of view, before becoming interested in 
submitting a project proposal - needed substantial support to turn a vague 
interest in "working on their teaching" into a concrete project idea which 
they could state in a project proposal. 

• The very contextual and locally grounded nature of the projects led to a 
flower garden of initiatives. Teachers enjoyed sharing their experiences, 
but working on different sites and starting with different problems made 
joint efforts for evaluation across contexts almost impossible. Thus, in 
most schools local teams designed their own instruments for assessment 
addressing their very specific goals. In general, that led to a multitude of 
small, locally significant findings, which addressed different aspects of 
enjoying, learning and doing mathematics across different age levels and 
school types. Taken together with the national context, which does not 
require state wide centralized tests at any time during schooling (which 
could for example measure achievement gain scores), it was not possible 
to make coherent statements on the overall impact on learning of the fund 
without a further massive intervention into those classrooms. Instead 
IMST had to content itself with measures of attitude, self-confidence, and 
subject-related anxiety. In these measures, IMST classes on average 
performed significantly better than the average Austria mathematics class 
(of that age group and school type; Andreitz, Hanfstingl, & Miiller, 2007). 
However, the participating teachers usually demonstrated high levels of 
job motivation and interest already when entering the projects, thus the 
good results may be due to the special self-selected group of teachers. 

• Each project had to hand in a report at the end of the year. Reports can 
have various purposes, which might at times conflict with each other. The 
report should present the project to the public, at the same time, writing is 
a means to engage in reflection. In general, teachers reported that this is 
the most difficult step of the entire project. Teachers (in Austria) are not 
used to write, reflective accounts of their classroom practices or projects. 
Retrospectively, teachers uniformly valued the experience as a vital piece 
of their learning from the project (see also Schuster, 2008). 

• Benke (2004) found a marked difference between those project reports (of 
the precursor of the IMST project, IMST 2 ) which were written by 
individuals and those which were written by teams - the latter included 
almost no reflective elements; teams tended to report on the results of 
reflection, but did not mirror the process of reflecting in writing. 
Moreover, the project reports of teachers reflect the value judgements and 
judgements of relevance of the writing teachers. These may at time be 
quite at odds with judgements of relevance of researchers or educators. 



• An open issue is the further use of the project reports. The reports are all 
published on the webpage for all projects. Teachers and their school enjoy 
having the project reports for internal and external communication 
purposes. Yet teachers are not prone to look up, what someone else has 
done or to learn about another project. IMST is presently working on a 
strategy to better disseminate the project reports to interested teachers - 
which is incidentally an issue action research, as a strategy for 
professional teacher development still needs to take up. Even if problems 
are locally grounded, and each community of inquiry needs to find their 
own answers to their own problems, the answers of the others make us 
richer, and we need to learn from them as well. 

Czech Republic: A Comenius Project on Understanding of Mathematics Classroom 

The intervention into professional teacher education reported here was developed 
within the scope of a more encompassing international project on "Understanding 
of mathematics classroom culture in different countries". Within this project, it was 
decided to collect video records of teaching episodes. Although originally not 
intended, these episodes became core elements of joint reflection and development 
of the participating teachers in the national and international meetings. Instead of 
videotaping a "regular classroom teacher", the project team began to carefully 
discuss and then plan individual lessons which covered the required curriculum. 
Thus, lessons turned into "teaching experiments" for all participants. The topic that 
was selected jointly for such an inquiry approach was one of the most difficult 
concepts in mathematics education at primary school level - the concept of 
fractions. 12 Thus, the Czech team agreed that the experimental teaching should 
focus on: (a) the creation of the notion of the part and the whole, and (b) the 
continuous enrichment of various modes of representation and interpretation. 

The cooperation in the Czech team gradually settled roughly on the following 

• Preparation of the teaching experiments usually began at a joint meeting of 
the Czech team of teachers and researchers. The group discussed the topic of 
the upcoming experiment, the potentialities of the use of various methods 
and techniques, and the mathematical content in greater detail if necessary. 
The topic of the experiment usually addressed the needs of the teacher who 
would then conduct the experiment in his or her class. 

• The teaching experiment was usually realized by one teacher in her class. If 
two of the teachers were teaching the same grade, the experiment was carried 
out by both of them. After the joint session, the preparation and lesson 
planning of the teachers was individual. 

12 Many teachers tend to use a schematic approach and so they focus on drills of numerical operations 
with fractions, and students usually master these operations relatively quickly. But, if we investigate the 
level of students' conceptions of fractions, we often find out that it is very low (Ticha, 2003). 



• The experimental teaching was video recorded by one researcher using a 
camcorder. During the project, the researchers made 25 video recordings of 
lessons (or parts of it). The intrusion into the course of the lesson was 
minimal as the students became accustomed to being video recorded. 

• The teacher who had taught the recorded lesson was the first to watch the 
recording and to select interesting segments which to discuss with 
colleagues. She consulted with the researcher who made the video recording 
about her choice of episodes before sharing with her colleagues. 

• The selected video episodes were (sometimes repeatedly) watched as a group 
but reflected upon individually by all members of the team. Therefore, all the 
members of the team were prepared for subsequent joint reflection. 

• The selected video episodes formed the basis for joint reflection of the whole 
team. These teacher discussions were usually recorded by the researchers, 
which enabled them to conduct a follow-up analysis of the joint reflection. 

• Several episodes were also jointly reflected by the international team. 
Throughout the project, the teachers, who taught at different grades levels, 

gradually began to change their perspective on the meaning and essence of 
mathematics education; in addition the researchers found their own understanding 
of the school practice was also changing. In joint reflections on these short video 
episodes, the teachers and researchers discussed, for example, possible approaches 
to teaching a certain topic and sources of various beliefs. After some time of close 
cooperation, the teachers began to recognize their different teaching styles and 
philosophies of school mathematics, didactical approaches, subject matter 
knowledge and so on (Bromme, 1 994) as fundamental sources of differences; and 
they used these insights when reflecting on their different professional orientation 
and undergraduate training. The discussion usually centred on two areas: (i) 
students' work, and (ii) teachers' opinions on how to approach a specific topic. 
What was surprising to the researchers was that the whole discussion was 
penetrated by considerations on the essence and meaning of mathematics education 
and also by considerations on the sense of joint reflections. They concluded that 
joint reflection should be used in particular for: (a) cultivation of the teachers' 
behaviour and actions, (b) formation of more perceptive teachers' approaches to 
students' ways of thinking and the ability to utilize them in teaching, and (c) 
becoming more conscious of moments valuable from the point of view of students' 
cognitive processes. However, teachers still needed some guidance on how to 
reflect on pedagogical situations (Scherer, Sobeke, & Steinbring, 2004). The 
project team recommends that prospective teachers should be systematically 
prepared for reflections on teaching and should be familiar with the relevant 
literature. Practising teachers (participating in the project) themselves admitted that 
they would appreciate some instruction and guidelines that they could follow in 
order to have more productive reflections. However, they would not like them to be 
too binding and admitted they were not really sure they would make use of these 



Using video recordings made it possible to create and cultivate skills in 
assessment of students' answers, to diagnose students' mistakes and their sources. 
Given the support of joint reflection, participants learned to exert deeper insight 
into the content taught and into how content was grasped by the students. Although 
the aim of the project was not to support professional development of the 
participating teachers, significant changes in their perception of teaching of 
mathematics could be observed, as well as changes of the teachers' reflections 
(Ticha & HoSpesova, 2006), and the teaching itself. We have to stress that the 
changes of the teachers differed depending on their personalities, education, 
experience, age and so on. We illustrate this idea with the following brief accounts 
of the change made by two teachers participating in the project, Anna and Cecily: 

Anna was entering the project with the stated intention to change her teaching of 
mathematics. The video recordings of her lessons taken at the beginning, reveal her 
considerable mastery of methodology and her effort to prepare for students such 
problems that they would be able to solve them without greater difficulties. 
However, she always tried to ensure that her students would also understand the 
process of numerical operations that they were taught. 

After a few meetings of our team, Anna stopped looking for merely effective 
methodological approaches. In her lessons, she began to create problem situations 
for which the solution would promote understanding and lead to building of 
concepts. For example, she found her inspiration in German textbooks and 
prepared the teaching experiment for her second grade students dealing with 
nonstandard procedures of subtraction. She asked her students to decide which of a 
set of given calculations were correct and also to explain why. As for the 
justifications of the students, she was not satisfied merely with calculation of the 
result. She insisted that the explanation should be comprehensible to all other 
students. The goal of teaching in the second grade is usually a practice of addition 
and subtraction of numbers up to 100. It is by no means usual to ask the students to 
develop their own procedures or to explain unusual procedures. In the discussion 
on the issue she stressed that she wanted to promote understanding of the counting 
procedures and realized how her students would cope with the problem. In the joint 
reflection, she stressed that she had no experience with such a type of problem; and 
her satisfaction with the students' performance was visible; the lesson is analysed 
in greater detail in Ticha and HoSpesova (2006). After that experience, Anna 
returned to the problem several times, modified it and followed her pupil's 
development. Her action research served her actual practical needs. 

Our second example tells of Cecily, who entered the project as a regular teacher 
with a solid knowledge about teaching and mathematics and who was interested in 
learning more about different approaches and representations and in further 
developing of her own abilities. For her, participation in discussions was a stimulus 
for further studies. Gradually, her interest moved from the effort to "show 
something", to a search for problems, misunderstandings and their sources and 
causes. She started to propose content and methods of investigations, sometimes to 
suggest modifications of the conditions of performed research. For example, she 
carried out her own instruction experiment with an entire class. Immediately after 



the lesson, she suggested a repetition of this experiment with a group of six 
students, which allowed studying individual processes. Another time, she 
suggested a modification of the research performed with teachers for her students. 
After a reflection of this teaching episode, she suggested to jointly reflect on the 
same episode with other participants (Ticha & HoSpesova, 2006). During the 
cooperation, Cecily entered her PhD study. 

Cecily constitutes an example for the claim that those teachers who are strongly 
invested in action research or reflective inquiry are prone to go on for further study 
to become a "true researcher" - in which they might leave the non-academic 
teaching profession (this has not happened in this case - yet). For this teacher, the 
participation and collaboration in the Comenius team meant a great challenge for 
her further education (Ticha & HoSpesova, 2006). This is also attested in her 
written self-reflections. 

After the first year of the project, Cecily wrote: 

Before starting the project I didn't know what to expect at all. I just knew that 
it was something about mathematics. 1 wasn't able to imagine the reason or 
the aim of the project. I told myself that I would see. 

After three years, she commented: 

[...] for me it was very interesting to meet teachers from other countries and 
compare our teaching experiences and opinions. I started to think about what 
we should have to prepare for our presentation. [...]. I found that it was very 
useful to see myself as another person during discussions with my colleagues 
because it is one of the ways of improving my work and thinking about 
myself differently. 

And after beginning her doctoral degree programme, she reflected: 

When watching the video, it is not easy to separate the outer and inner view 
of my activity during the lesson (compare this with the above). I think that it 
is impossible to see oneself as another person. When watching the recorded 
lesson, I relive the whole lesson step by step again and again. I can see what I 
could not see during the lesson itself, I can find out why the students 
misunderstood, but sometimes, despite all my effort, I do not know why 
something happened the way it happened and why some misunderstandings 
both of the students and myself could not be solved. I am not sure whether 
my conjecture is correct; it seems probable to me. It is really useful in this 
position to have a view of another person who can see the matter in a 
different way, and the subsequent discussion can lead to finding the solution. 
When analysing our own lessons and behaviour, we can discover many 
things. At certain moments, I might think that there is nothing more to 
analyse, but the view of other people and the discussion with them shows that 
there are matters that I haven't noticed. 





Looking back, we find ourselves still very much at the beginning. Practitioners and 
educational researchers working with action research are generally positive about 
their experiences, even though the objectives of researchers and teachers seemingly 
so similar - to improve teaching and achieve higher standard of education - are 
vastly different. In general, researchers look for answers to theoretical questions 
while teachers deal with practical problems. For teachers doing action research, it 
brings about a change of understanding and assessing of (a) their own role, (b) 
what is essential for their work, (c) their own professional knowledge, and it brings 
about (d) a new way of reflecting their own practice. To researchers, it means (a) a 
deepening of understanding of processes, which take place at mathematics 
teaching, (b) improvement of quality of teacher competence assessment (from 
intuition to junior researcher), (c) possibility to influence teacher competences, and 
(d) improvement of didactic research. 

But as reflected above, the strong and theoretically well supported tenet to work 
with problems located at the chosen sites, precludes any easy systematic evaluation 
of the overall impact of action research and the differential contribution of specific 
implementations. Moreover, the practice-theory problem, that is the question of 
how to communicate insights from practice and insights from educational research 
at large - in particular the community of mathematics didactitians - between the 
communities of practitioners and researchers, is very much at the heart of effective 
action research. 

In brief, action research remains a promising strand of professional development 
and education research, which still has to prove itself. However, regardless of the 
specific contributions of action research to professional development, engaging in 
action research forces the educational researcher to consider many questions very 
much at the heart of educational research for teacher development and learning in 
general; for example: How can we examine changes in teachers' beliefs and 
knowledge? How can university people support teachers to reflect more deeply? 
What criteria are characteristic for a "reflective teacher"? A teacher has different 
tensions and priorities than a researcher. How does a teacher-researcher balance 
these tensions? How can we examine the benefit of joint reflection? And ultimately 
following Dewey's vision: How can we support teachers' growth to become truer 
to themselves as teachers and persons? 


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Gertraitd Benke 

Institute of Instructional and School Development 

University ofKlagenfurt 


Alena Hospesovd 

Faculty of Education 

University of South Bohemia Ceske Budejovice 

Czech Republic 

Marie Tichd 

Institute of Mathematics, v. v. i. 

Academy of Sciences of the Czech Republic 

Czech Republic 






Teachers and Didacticians in Collaboration 

Teachers and didacticians both bring areas of expertise, forms of knowing and 
relevant experience to collaboration in mathematics teaching development. The 
notion of inquiry community, provides a theoretical and practical foundation for 
development. Within an inquiry community all participants are researchers (taking 
a broad definition). With reference to a research and development project in 
Norway (Learning Communities in Mathematics - LCM) this chapter explains the 
theoretical notions, discusses how one community was conceived and emerged in 
practice and addresses the issues contingent on emergence and sustaining of 
inquiry practices. In doing so it provides examples of collaborative activity and the 
reciprocal forms of expertise, knowing and experience that have contributed to 
community building. It illuminates issues and tensions that have been central to the 
developmental process and shows haw an activity theory analysis can help to 
navigate the complexity in characterizing development. 


This chapter focuses on co-learning inquiry, a mode of developmental research in 
which knowledge and practice develop through the inquiry activity of the people 
engaged (Jaworski, 2004a, 2006). This involves the creation of inquiry 
communities between didacticians and teachers to explore ways of improving 
learning environments for students in mathematics classrooms. Research both 
charts the developmental process and is a tool for development. The chapter draws 
on a research and development project in Norway, 1 Learning Communities in 
Mathematics (LCM), for which co-learning inquiry and communities of practice 
have formed a theoretical basis. The nature of inquiry, development and research in 
the project is used as a basis for extracting more general principles and issues. 

The LCM project focused on how learners of mathematics at any level of 
schooling can develop conceptual understanding of mathematics that is reflected in 
nationally and internationally measured success. The project was rooted in 

' The LCM project was funded by the Research Council of Norway (RCN) in their advertised 
programme Kunnskap, Utdanning og Laering (Knowledge, Education and Learning - KUL): Project 
number 157949/S20. 

K. Krainer and T. Wood (eds.). Participants in Mathematics Teacher Education, 309-330. 
© 2008 Sense Publishers. All rights reserved. 


established systems and communities in which education is formalised and 
mathematics learning and teaching take place. 2 

The chapter weaves theory and practice to address meanings and roots of co- 
learning inquiry and inquiry community and issues in creating and sustaining 
inquiry communities for development of learning and teaching mathematics. 


Knowledge in Sociocultural Settings 

Knowledge is seen to be both brought by people engaged in the educational 
process and embedded in the practices and ways of being of these people - 
students in classrooms, teachers of mathematics in schools, and mathematics 
didacticians in a university. 

According to Lave and Wenger (1991), knowledge is in participation in the 
practice or activity, and not in the individual consciousness of the participants. 
"The unit of analysis is thus not the individual, nor the environment, but a relation 
between the two" (Nardi, 1996, p. 71). So, the practice, or activity, in which 
participants engage is crucial to a situated (social practice theory) perspective. 
Wenger (1998) talks of belonging to a community of practice involving 
engagement, imagination and alignment. The terms participation, belonging, 
engagement and alignment all point towards the situatedness of activity and the 
growth of knowledge in practice. 

Within the communities of our project we recognize both individuals and 
groups: that is we ascribe identity to both. Holland, Lachicotte, Skinner, and Cain 
(1998, p. 5) write, "Identity is a concept that figuratively combines the intimate or 
personal world with the collective space of cultural forms and social relations". 
Identity refers to ways of being (Holland et al., 1998). We talk about ways of being 
in the LCM project community and in the other various communities of which 
project members are a part, leading to a concept of inquiry as a way of being 
(Jaworski, 2004a). Inquiry is first of all a tool used by participants in a community 
of practice in consideration and development of the practice, that of mathematics 
learning and teaching in classrooms. Inquiry mediates between the activity of the 
classroom and the developmental goals of participants. Participants engage in 
action that involves inquiry and learn from the outcomes of their action relative to 
established ways of being. Relationships between individuals and the communities 
in which they are participants are complex with respect to the forms of knowledge 
they encompass and growth of knowledge within the communities. 

Wertsch (1991, p. 12) emphasises that "the relationship between action and 
mediational means is so fundamental that it is more appropriate, when referring to 
the agent involved, to speak of 'individual(s)-acting-with-mediational-means' than 

2 A copy of the project proposal can be obtained from the author by direct communication. 



to speak simply of 'individual(s)'". Wertsch refers to Vygotsky's (1978, p. 57; 
emphasis in original) well known law of cultural development which states: 

Every function of a child's cultural development appears twice: first, on the 
social level, and later, on the individual level; first between people 
(interpsychologicat), and then inside the child (intrapsychological). This 
applies equally to voluntary attention, to logical memory, and to the 
formation of concepts. All the higher functions originate as actual relations 
between human individuals. 

Such a perspective sees learning as participation in social practice or activity. 
As we participate we "take part" in the practices or activities involved, grow into 
those practices or activities, and learn through our doing and acting. We engage 
mentally and physically, and communicate with those around us. We use the 
language, words or gestures, of the practice or activity to engage and communicate. 
Different social groups use language in different ways and within any group we 
speak or learn to speak the group language. 

Leont'ev (1979, pp. 47-^8) writes, 

in a society, humans do not simply find external conditions to which they 
must adapt their activity. Rather these social conditions bear with them the 
motives and goals of their activity, its means and modes. In a word, society 
produces the activity of the individuals it forms. 

Thus, activity is necessarily motivated; actions have explicit goals, and 
individuals engage in activity with goal-directed action leading to integral 
formation of "the intramental plane". Mediation is central to this formation, with 
the mediational means (tools, signs or other) a key focus in activity theory 
(Leont'ev, 1979; Wertsch, 1991). 

Thus, starting from identity as meaning belonging in practice, with knowledge 
firmly rooted in practice (Wenger, 1998), we move to identity as the mediational 
formation of the intramental plane through goal-directed action (Wertsch, 1991). 
This extension of belonging through goal-directed action offers a theoretical 
grounding for the extension of alignment to critical alignment through processes of 
inquiry. I shall return to this below. 

Co-Learning Inquiry 

Co-learning inquiry means people learning together through inquiry; inquiry being 
a mediational tool as indicated above. The term "co-learning" comes from Wagner 
(1997, p. 16) who writes 

In a co-learning agreement, researchers and practitioners are both participants 
in processes of education and systems of schooling. Both are engaged in 
action and reflection. By working together, each might learn something about 
the world of the other. Of equal importance, however, each may learn 



something more about his or her own world and its connections to institutions 
and schooling. 

An aim of the LCM project was that didacticians from the university and 
teachers from schools would work together to explore and develop mathematics 
learning and teaching in classrooms. In such collaboration, both groups are 
practitioners and, since both engage in exploration and inquiry, both are 
researchers. We thus adapted slightly the words from Wagner (1997, p. 16) to read: 
"teachers and didacticians are both practitioners and researchers in processes of 
education and systems of schooling". The simple aim, that didacticians and 
teachers would work together as both practitioners and researchers, was both a 
guiding force for LCM and a source of tension in relation to power and hierarchy. 
Didacticians conceptualized the project, gained the funding, invited participation 
from schools, and set up the basic project design. Given such clear "ownership" of 
the project, could it be possible to redress the obvious hierarchy and create some 
kind of sharing of power and responsibility? This question will be addressed 
throughout the chapter with relation to the developmental project (LCM) and the 
theoretical perspectives outlined above. 

Inquiry Community 

Inquiry community was part of didacticians' vision for the LCM project; a 
theoretical concept rooted in wide previous experience and a number of key 
sources. According to Chambers Dictionary, inquiry means to ask a question; to 
make an investigation; to acquire information; to search for knowledge. Wells 
(1999, p. 122) speaks of "dialogic inquiry" as "a willingness to wonder, to ask 
questions, and to seek to understand by collaborating with others in the attempt to 
make answers to them". He emphasizes the importance of dialogue to the inquiry 
process in which questioning, exploring, investigating, and researching are key 
activities or roles of teachers and didacticians (and ultimately students). These 
activities can be discerned through the analysis of dialogue in interactions within 
the community. 

Didacticians had distinguished between use of inquiry as a tool in teaching and 
learning, and developing inquiry as a way of being, so that the identity of an 
individual or group within an inquiry community would be rooted in inquiry 
(Jaworski, 2004a). Developing inquiry as a way of being involves becoming, or 
taking the role of, an inquirer; becoming a person who questions, explores, 
investigates and researches within everyday, normal practice. The vision has much 
in common with what Cochran-Smith and Lytle (1999) speak of as "inquiry as 
stance" - the stance of teachers who engage in an inquiry way of being. 
Participants in a community of inquiry aspire to develop an inquiry way of being, 
an inquiry identity, in engagement in practice. A focus of the LCM project was to 
explore what inquiry could mean in mathematics classrooms and in the activity of 
teachers and didacticians trying to explore development of mathematics teaching 
and learning. 



These words suggest that we do not necessarily have inquiry ways of being in 
"normal" practice. Brown and Mclntyre (1993), researching teaching in classrooms 
from observation of classroom activity and interviews with teachers, suggested that 
teaching and (earning in classrooms develops "normal desirable states". Teachers 
and students find ways of working together that fit as well as possible with 
expectations of educational and social systems and groups and allow a workable 
environment. The workable environment comes from an implicit agreement 
between teachers and students about what is expected, and what is acceptable in 
classroom activity - a sort of didactic contract (Brousseau, 1984). Such ways of 
working and being in classrooms might be characterized as communities of 
practice 3 (Lave & Wenger, 1991), in which participants align themselves with the 
normal desirable state. However, the normal desirable state does not necessarily 
foster the kinds of mathematical achievement didacticians, and society more 
broadly, would like to see. 4 

In terms of Wenger's (1998) theory, that belonging to a community of practice 
involves engagement, imagination and alignment, we might see the normal 
desirable state as engaging students and teachers in forms of practice and ways of 
being in practice with which they align their actions and conform to expectations. 
Imagination ensures comfortable existence within the broader social expectations 
and acceptable or desirable patterns of activity. 

One of the reasons for introducing inquiry as a tool - for example, in designing 
inquiry tasks to stimulate inquiry in the classroom - is to challenge the normal 
(desirable) state and question what it is achieving. For example, if students are 
learning mathematics through text book exercises, in which the goal is to practise 
skills and become fluent with operations, we might ask questions about the degree 
of conceptual understanding that is afforded by this practice. If the normal 
desirable state is to be sure that students can do what is required, and not to worry 
too much about understanding, then it could be that we are denying students an 
important opportunity - to understand the mathematics they are learning, and to 
relate particular ideas more widely, both in mathematics and in real world 
applications. So, we might ask, what can we do in classrooms to enable students to 
understand better the mathematics they meet in text book exercises? This is a 
developmental question. As soon as we strive to address such a question, we enter 
an inquiry or a research process. 

In an inquiry community, we are not satisfied with the normal (desirable) state, 
but we approach our practice with a questioning attitude, not to change everything 
overnight, but to start to explore what else is possible; to wonder, to ask questions, 

3 The practice is that of engaging in classroom activity according to the norms and expectations of the 
particular setting in which activity takes place. Such practice is often referred to as mathematics 
teaching and/or teaming. 

4 The TIMSS and PISA studies provide ample evidence of this, for Norway and for many other 
countries. See, for example, Kjaernsli, Lie, Olsen, and Turmo, 2004; Gronmo, Bergem, Kjaernsli, Lie, 
and Turmo (2004); Mullis, Martin, Gonzalez, and Chrostowski (2004); Mullis, Martin, Beaton, 
Gonzalez, Kelly, and Smith ( 1 998). 



and to seek to understand by collaborating with others in the attempt to provide 
answers to them (Wells, 1999). In this activity, if our questioning is systematic and 
we set out purposefully to inquire into our practices, we become researchers. 

The community of the LCM project, set up to generate a community of inquiry, 
had to learn, to grow into, to come to know what it could mean to work in inquiry 
ways, to develop questioning attitudes, to design inquiry tasks and to foster 
students' own inquiry. Thus the community of inquiry was an emergent rather than 
an established form of practice. Inquiry practices in schools bring new elements to 
established practices. Thus, in order to move from a community of practice to a 
community of inquiry, participants will engage in existing practices, aligning to 
some extent with those practices, but in a questioning or inquiry mode. This has 
been termed "critical alignment" (Jaworski, 2006). It involves a recognition that 
within existing practices, alignment (in Wenger's terms) is essential, but if we 
bring a critical attitude to alignment - that is we question, we explore, we seek 
alternatives while engaging - then we have possibilities to develop and change the 
normal states. 

Activity Theory as an Analytical Tool 

The theoretical ideas outlined above have allowed us to conceptualise the roots of 
inquiry communities; we have found, however, that they do not go far enough in 
allowing us to analyse the various forms of data we have generated in order to cut 
through complexities in the various communities in which the LCM project has 
been embedded. For this reason we have turned to activity theory which has 
allowed us to inter-relate concepts of community, inquiry and critical alignment in 
seeking to explain issues and tensions in the project and emergent growth of 

We start here from transitions between intermental and intramental planes 
(Vygotsky, 1978; Wertsch, 1991) and the roles of didacticians and teachers in 
promoting development in mathematics classrooms. As I shall explain below, 
practices within the LCM project, although goal-directed, were not pre-designated. 
It is one thing to propose creation of a community of inquiry and quite another to 
realize it. A major part of our developmental activity and associated research 
involved exploring the creation and nature of an inquiry community. The inquiry 
community was emergent in the project as were the knowledge and learning 
associated with it. As the people of the project engaged with the activity of the 
project within the project community, also working simultaneously in other 
communities of practice (schools or university), people learned and knowledge 
grew. From knowledge and activity within existing communities of practice, and 
activity within the project, new understandings, and new ways of being and acting, 
emerged. In Wertsch's (1991) terms, people acting with mediational means within 
their respective communities, with goals relating to developing mathematics 
learning and teaching in classrooms, form, as part of their communicative 
interaction, their /wtermental plane. We see the intermental plane to be the learning 
and knowing that occurs within the community as a whole, with the formation of 



intramental planes as individuals participate in mediated action. Leont'ev's (1979) 
concepts of motivated activity and goal-directed action have been employed in 
analysis of data to chart learning in LCM (Goodchild & Jaworski, 2005; Jaworski 
& Goodchild, 2006), along with EngestrSm's (1999) mediational triangle and 
concept of expansive learning (see below). 


Starting Points: From Motives to Goal-Directed Action 

The term "community" designates a group of people identifiable by who they are 
in terms of how they relate to each other, their common activities and ways of 
thinking, beliefs and values. Activities are likely to be explicit, whereas ways of 
thinking, beliefs and values are more implicit. Wenger (1998, p. 5) describes 
community as "a way of talking about the social configurations in which our 
enterprises are defined as worth pursuing and our participation is recognisable as 

According to Rogoff, Matusov, and White (1996, p. 388), in a learning 
community, "learning involves transformation of participation in collaborative 
endeavour". The idea of inquiry community makes the nature of transformation 
more explicit: didacticians and teachers (and ultimately students) will engage 
together in inquiry activity. What such activity should or could consist of, and how 
it should or could relate to activity in existing communities of practice, the 
classrooms, schools and university settings was a focus of research in LCM. 

LCM was motivated by developmental aims from which project activity was 
designed. In submitting a proposal to seek funding, didacticians proposed certain 
forms of action which would give shape to the project. These included workshops 
for teachers and didacticians in university settings, design of tasks for workshops 
and classrooms, teacher teams in schools for design of classroom activity, and 
collection of data from all activity. Thus, realization, or operationalization of the 
project required activity in which this design was implemented into project 
practice. We proposed engagement in an inquiry cycle (plan, act & observe, reflect 
and analyse, feedback) in the design process as the basis for our practical 
realization of a developmental research paradigm - more of this below. 

The nature of the inquiry cycle was something that emerged in project activity. 
The proposed practices set out in the initial design were what engaged us initially 
along with the philosophy of co-learning inquiry. We (didacticians) wished to 
collaborate with teachers as partners in developing and researching mathematics 
teaching in classrooms (Jaworski, 1999). We wanted to try to avoid positions of 
offering teachers models of practice and supporting their implementation, or of 
bringing teachers into developmental practice after the design stage and including 
them only then in the action (Jaworski, 2004b). Nevertheless, the project had been 
conceived by didacticians: the philosophical basis of the project (in co-learning 
inquiry) was not negotiable but was clearly open to interpretation; the more 
practical aspects of project design could be negotiated but award of funding 



brought with it a responsibility for didacticians to achieve what had been set out, so 
at some levels it was not possible to start to (re)negotiate the ground with teachers. 
So> the initial position was that motivation for the project was in place together 
with some designated action and goals. A major developmental question at this 
stage was how to bring teachers into the project. 

Action and Inquiry 

In creating an inquiry community, the participants have to come together in goal- 
directed action. Establishing goals within a community is itself a developmental 
task, and goes with initial action. In their invitation to schools and teachers to 
participate, didacticians set out the principles of the project and outlined its 
operation based on workshops at the university and innovation in schools and 
classrooms. Schools were recruited for two years, with the possibility of a third 
year. 5 

So, with regard to action and goals some things were taken as basic (e.g., 
workshops and co-learning inquiry) and (many) others were open to negotiation 
and experimentation in project activity. The motivating principle on which we all 
agreed (didacticians and teachers) was our desire to develop better learning 
environments for students in mathematics at the levels of schooling with which we 
were associated. Unsurprisingly, the ways of thinking about this principle were 
deeply related to the communities of practice from which we came, and these 
varied across the schools and between schools and university. The knowledge we 
brought to the project initially was also deeply embedded in our established 
communities with sociohistorical precedents and cultural practices forming 
identities in the project. 6 

Two examples from LCM illustrate the initial position. Didacticians' planning 
for workshops was rooted in their philosophy for the project and their knowledge 
of educational literature and research relating to teacher education and 
developmental practice in mathematics classrooms. Some had pioneered small 
group problem solving in mathematics teaching at the university (Borgersen, 1994) 
and all believed strongly in investigative approaches to teaching mathematics at 
any educational level. This embedded knowledge was highly motivational in the 
activity that didacticians planned for workshops. Teachers came from an 
educational system in which initial teacher education was provided in a university 
with practice in schools, and continuing teacher education was provided through 
workshops and seminars led by teachers from the university. There was 
expectation by at least some teachers that didacticians would lead the way in 
proposing developmental activity. A quotation from a teacher Agnes, in a focus 

5 Eight schools from early primary to upper secondary joined the project and 40 teachers participated 
during three years. Some funding was given to schools to support teachers in attending workshops in 
school time. Further details can be found in Jaworski (2005). 

6 Norwegian culture and society along with educational values and systems were in common for most 
project participants (teachers and didacticians). 



group interview at the end of two years of classroom activity indicates that she 
struggled in the beginning because didacticians did not seem to show teachers what 
to do. 

Agnes: [...] in the beginning I struggled, had a bit of a problem with this 
because then I thought very much about you should come and tell us how we 
should run the mathematics teaching. This was how I thought, you are the 
great teachers [...] (FG060313. Translated from the Norwegian by Espen 

Thus teachers found it difficult in the initial stages to initiate innovative activity 
in their schools because it had not been their custom to think of doing so and they 
expected a clear lead from didacticians. There were other barriers as well, which I 
shall come to. 

In LCM, particularly in the first year, it was the workshops which led the way in 
bringing participants together to build community and create an inquiry approach 
to thinking about mathematics, teaching and learning. Didacticians planned tasks to 
stimulate thinking and action: these were increasingly influenced over three years 
by teachers' comments, suggestions and requests for particular forms of activity. In 
any workshop, teachers and didacticians worked side by side on tasks, usually in 
small groups, in a mode intended to create genuine collaboration in doing 
mathematics and talking about associated classroom issues. Groups were organized 
sometimes to cross school levels, at other times to align with levels more or less 
finely. Plenary sessions allowed input on relevant topics, presentations from school 
activity (often using video recordings from classrooms), and feedback from group 
activity and discussion. 

In all workshops, mathematical tasks were chosen or designed carefully (mainly 
by didacticians - discussed further below) for their mathematical or didactical 
appropriateness for the stage of the project. In the first workshops, problems were 
chosen which had rich potential for stimulating mathematical thinking and which 
were accessible to people with widely different mathematical experience. Later, 
problems or tasks were designed related to curriculum topics. All work on tasks led 
to discussions, in both small group and plenary, around the didactics and pedagogy 
of creating tasks for classrooms and the associated issues. 7 

The workshops were spaced throughout the school year so that, between 
workshops, school activity and innovation could take place. Two forms of activity 
in school emerged from this opportunity. In some cases, teachers took tasks from 
the workshops and, with suitable modification, used them in their classrooms with 
students. Frequently, reporting at a workshop included presentations from such 
student activity. In other cases, the teacher team in a school designed a task or set 
of tasks to bring an inquiry approach to a curriculum topic. Varying degrees of 
collaboration between teachers and didacticians were involved in designing and 

A special issue of the Journal of Mathematics Teacher Education (JMTE 4-6, 2007) is devoted to 
research into the design and use of mathematical related tasks in teacher education. 



planning such tasks. Didacticians often recorded classroom activity on video when 
such innovation took place, and video became an important medium in the project 
for sharing experience of task design and use in schools. We see here clear 
examples of mediation between inter- and intra-mental planes. 

Thus early action took place in workshops in the university and in school 
activity stimulated by the workshops. Inquiry was evident in the planning process, 
in ways in which teachers took workshop ideas back to schools and tried out ideas 
in classrooms and in the developing relationships between the participants as 
activity progressed. I shall talk later about the outcomes of such activity in terms of 
participants' learning and issues and tensions which arose. 

An Inquiry Cycle in the Design Process 

From the beginning of the projects, design was a central factor in creating 
workshop or classroom activity and innovation. Didacticians followed loosely a 
design research approach to creating activity in workshops (Kelly, 2003; Wood & 
Berry, 2003). The approach was inquiry-based and iterative (plan, act and observe, 
reflect and analyse, feedback to planning) and was in Kelly's terms "generative and 
transformative" (2003, p. 3). Typically, following an initial planning meeting, a 
small team of didacticians took on the design of tasks according to agreed criteria 
for the coming workshop: for example, tasks relating to algebra at a range of levels 
including opportunities for generalization and justification of conjectures. The 
small team circulated the outcomes of their design process and these were 
discussed in a subsequent meeting. After the workshop, one meeting of 
didacticians was dedicated to reflecting on the workshop activity including 
outcomes from the use of tasks; these reflections feeding into subsequent decision 
making and planning. This inquiry process was centrally important in sharing 
knowledge and expertise among didacticians, stimulating creativity, generating a 
group outcome in terms of tasks for a workshop, and building new knowledge 
within the didactician community. 

Didacticians envisaged a similar process for school teams planning for the 
classroom. Unsurprisingly, the outcomes were very variable, and related to 
particular school circumstances. While, in at least one school, the design cycle, in 
planning and implementing tasks and reflecting on their use by students, was 
exemplary (Fuglestad, Goodchild, & Jaworski, 2006; Hundeland, Erfjord, 
Grevholm, & Breiteig, 2007) in other schools planning was more ad hoc, often 
individual and relating to one class only (Daland, 2007). The most common 
practice observed was teachers' use of workshop tasks, modified for classrooms. 
Teachers reported in subsequent workshops from their classroom activity and the 
engagement of their students, and video extracts showed evidence of classroom 
innovation. The words of Agnes, continuing from the quotation above, testify to 
teacher growth through this process: 

[...] but now I see that my view has gradually changed because I see that you 
are participants in this as much as we are even though it is you that organise. 



Nevertheless I experience that you are participating and are just as interested 
as we are to solve the tasks on our level and find possibilities, find tasks, that 
may be appropriate for the students, and that I think is very nice. So I have 
changed my view during this time. (FG_060313. Translated from the 
Norwegian by Espen Daland) 

Project activity in schools proved a major learning experience for didacticians as 
I discuss below. 

A Developmental Research Paradigm 

The central use of design, as in design of tasks and activity for workshops and 
classrooms suggested a design research approach to the project. Didacticians would 
design for workshops, albeit taking into account strongly the views and suggestions 
of teachers. Teachers would design for the classroom, drawing on experiences in 
workshops and inviting didacticians' contributions as appropriate. However, the 
theory of design research (see Wood & Berry, 2003) proved too "clinical". The 
design cycle, even in the activity of didacticians, was rarely conceived "up front", 
and emerged largely from human interactivity around the aims of the project. In 
schools, it was often hard to recognize clearly the elements of a cycle, intertwined 
as they were with the multitude of factors that make up teachers' lives. 

Here we recognize the developmental nature of the projects - activity emerged 
from engagement. Action, observation, reflection and analysis in the inquiry/design 
cycle led to growing awareness of the nature of co-learning in the projects. This 
inquiry cycle was overtly a learning process for all participants who acted as 
insider researchers, inquiring into their own practices and feeding back what they 
learned into future action (Bassey, 1995; Jaworski, 2004b; Goodchild, 2007). The 
systematic nature of such inquiry varied considerably across the project. 

From the beginning, didacticians collected data as far as possible from all 
activity - all meetings at which didacticians were present were recorded on audio, 
all workshop activity on video or audio, photographs were taken and documents 
carefully stored. Some school meetings were audio recorded and some classroom 
activity video recorded. A large data bank was organized to which all didacticians 
had access. These data were not related to particular research questions; rather 
research questions evolved through activity and data was used according to need. 
As didacticians followed up initial research questions in analysis of data and 
writing of papers, more refined questions emerged which then fed into future 
activity and further research. In this way, the emergent nature of research in the 
project became centrally visible, and it was possible to trace links between research 
activity and developmental progress. 8 

See Gravemeijer (1994) and Goodchild (Volume 4 of this Handbook) for related and extended 
accounts of developmental research. 




The essence of an inquiry community is that, through goal-directed action in 
communities of practice, participants explore, inquire into, their own practice with 
the motive of learning how to improve the practice (see also Benke, HoSpesova, & 
Ticha, this volume). All participants engaged in the project community, but they 
were also a part of other communities which made demands on their work and 
lives, and the inquiry process resulted in a more critical scrutiny of the range of 
practices and possibilities they afforded. Thus, didacticians and teachers, in their 
respective established communities both aligned with the practices of those 
communities and looked critically at their engagement. Teachers participated in the 
day to day life of their schools and, integrally, explored the use of inquiry-based 
tasks in their classrooms and observed their students' mathematical activity and 
learning. Didacticians collected and analysed data and wrote research papers, as 
expected of university academics and, integrally, explored the design of tasks for 
workshops and their work with teachers in school environments to support teachers 
in their project activity. Activity in the project community emerged from action. 
Action in the form of task design led to action in a workshop, which led in its turn 
to action in schools, each of these feeding back to inform succeeding stages of 
activity. The inquiry community of the project could not be separated from the 
established communities of which project members were a part. Interaction 
between established communities, their joint enterprise, mutual engagement and 
shared repertoire (Eriksen 2007; Wenger, 1998), and the emerging project 
community led to recognition of a complexity of inter-relations, issues and tensions 
as the project progressed. As indicated earlier, didacticians used activity theory to 
try to make sense of the complexity and address issues and tensions. 

Mediated Action and Engestrom 's Triangle 

Relating to Vygotsky's (1978) law of cultural development and ideas from 
Leont'ev (1979) and Wertsch (1991), expressed above, a simple triangle (see 
Figure 1) expresses the mediational process as individuals or groups (the subject of 
activity) engage in action to achieve goals (the object of activity). 

In all cases, according to the theory, activity is mediated: for example, activity 
in workshops is mediated by the tasks in which teachers and didacticians engage; 
activity in schools is mediated by the ideas teachers' bring from workshops related 
to tasks for the classroom and approaches to working with students. 

However, according to Engestrom (1998) this "simple" mediational triangle 
ignores the "hidden curriculum", the factors in education in schools that influence 
fundamentally what is possible for teachers and their students, and ultimately for 
didacticians in a developmental project such as LCM. Engestrom (1998, 1999) 
extended the simple triangle to the more complex version (see Figure 2). 







Based on Vygotsky's model of a complex mediated act 

Figure 1. The simple mediational triangle. 



4 OBJECT | — K 





Engestrom's 'complex model of an activity system' 

Figure 2. Engestrdm 's mediational triangle including the hidden curriculum. 

The progression from subject to object can be achieved in mediation through 
any of the paths indicated. Rules include the curriculum and its assessment, the 
ways in which school and educational systems operate, the societal and political 



expectations of schools and teachers. Community includes the established 
communities discussed above, as well as the project community in LCM. Division 
of labour includes the roles of participants, teachers in their school system, 
didacticians within a university setting; new roles developing through the project. 

Issues and tensions arise when elements of the hidden curriculum challenge the 
achievement of goals. I use the theory of mediated action within communities of 
inquiry and the hidden curriculum expressed in Engestrom's triangle to present 
some of the outcomes of the LCM project (in the rest of this section) and lead to 
more general observations concerning development in communities of inquiry (in 
the final section of this chapter). 

Didacticians ' Roles 

A tension with which didacticians have grappled since the beginning of the project 
concerns a didactician's role in working with teachers, either in a workshop or in a 
school environment. To what extent were we to offer our own thinking, viewpoint 
or expertise? In one early meeting, considering our role in a workshop small group, 
the term "coordinator" was used and rejected. Someone equated it with being "the 
boss". The words "facilitator" was preferred (Cestari, Daland, Eriksen, & Jaworski, 
2006). It was clear that the didactician in a group had some responsibility to ensure 
the smooth working of the group according to the declared task. This might mean 
ensuring that all participants were included in dialogue and activity. It might mean 
helping to keep the group focused. It might mean taking initiative to suggest roles 
for participants. It was agreed that it should not mean explaining the mathematics, 
or giving the solution of a problem. However, to what extent should a didactician 
participate in the mathematics? To what extent should he or she present a personal 
point of view in discussion? We had no clear answers to such questions. It 
remained for us to work according to broadly agreed principles and respond to 
particular circumstances. Activity was mediated through workshops tasks, 
experience from our activity in other communities, responses from the community 
of teachers present and so on. Mediation through subsequent sharing of experience 
with the didactician team enabled our awareness to grow and strengthened our 
ability to act knowledgeably according to agreed principles. For example, after one 
workshop, a didactician praised the actions of one colleague in enabling discussion 
in a small group. A further meeting was planned to watch a video recording of this 
group and to synthesise from the praised actions. From such interactivity over time, 
we learned both to live with uncertainty and to recognize the nature of growth in 
being a didactician in such a project. Despite saying these things so simply, this 
was not always a comfortable process. 

The Locus of Power and Control 

This issue of the didactician's role in activity with teachers adumbrates a 
fundamental issue that underpinned much of the LCM project - that of where the 
power and responsibility in the project was located, and its implications. 



Undeniably, the project originated with the didacticians; they were responsible to 
the research council, owned both conceptualization and operational isation to a high 
degree, and controlled funding. Schools had volunteered to be in the project and 
signed a contract with the university regarding their participation (Jaworski, 2005). 
Teachers participated with willingness and enthusiasm, and there was also much 
evidence of enjoyment. Teachers were also critical of what they experienced, and 
expressed points of view that were not always in accord with didacticians' 
concepts of events. 

For example, although workshop activity in small groups which crossed school 
levels was presented by didacticians as valuable for understanding students' 
experience beyond one's own level, teachers preferred overwhelmingly to work 
with colleagues at the same school level, and said so! After the very early months, 
small groups were usually same-level (and sometimes same-school) groups. Some 
teachers were critical of mathematical problems that were not clearly related to a 
topic in their own curriculum. They indicated that demands of curriculum and 
available time meant there was no possibility for them to use such problems, even 
though the problems were interesting and often fun to engage with. One teacher 
expressed this point of view after having chosen himself to engage with a 'fun' 
problem in a workshop. The implication was that in his lessons there was no time 
for 'fun'. Didacticians responded to such comments by designing mathematical 
tasks which could be seen as clearly curriculum-related, but nevertheless might be 
fun to engage with. Teachers responded that such tasks could be seen as valuable, 
but were much more time consuming than the text book tasks they used. However, 
the teachers expressing this point of view in one school invited didacticians to 
engage with them in designing more open tasks that could engage students 
conceptually. This resulted in a set of lessons, according to the teachers, quite 
different from those they held normally. They reported that students had seemed to 
have a better understanding of the mathematical concepts than earlier groups. 
Nevertheless, they were clear that they could not afford generally the amount of 
time demanded by these tasks (Fuglestad et al., 2006; Hundeland et al., 2007). 

We see here clear examples of critical alignment by both didacticians and 
teachers - a complex set of actions and reactions in and to project activity closely 
related to school activity. On the one hand, activity was led by design of tasks and 
group organization designed by didacticians. On the other, teachers' responses and 
perspectives led to reconceptualization and redesign; for example, groups became 
mainly same-level groups; tasks were increasingly curriculum-related. Teachers 
spoke from their own experiences and perspectives rooted in their normal activity 
in school communities and from the demands of the rules of schooling, for 
example the pressure of needing to "cover" the national curriculum. Rules and 
communities mediating the thinking and actions of teachers impinged on the 
project and mediated the design of tasks and workshop groupings. In order to 
achieve project goals didacticians needed to recognize and respond to teachers' 
concerns. Teachers surprised didacticians nevertheless by engaging in activity in 
ways that showed workshop goals being achieved in classrooms. Thus, control 



shifted between didacticians and teachers in interesting ways showing a complex 
division of labour in the project. 

Mutual Adaptation and Learning 

> 9 

The first year of activity with schools constituted Phase 1 of the project. Before 
the start of the second year (Phase 2), didacticians responded to teachers' 
comments on workshops by holding a consultative meeting. Teachers were invited 
to express frankly their views on workshops and to make suggestions for 
workshops in the coming phase of activity. Many indicated that finding time in 
school for the kinds of planning meetings they needed to design activity for 
classrooms was extremely difficult. School structures militated against such 
meetings and time was limited. They would like the opportunity to plan together 
with colleagues from other schools at the same level, to produce classroom tasks 
and to report on the classroom activity on a future occasion. These suggestions 
were so strongly supported across school levels, that Phase 2 of the project became 
structured accordingly. The Norwegian phrase "planlegge et opplegg" (devise the 
lesson plan) became a watchword for Phase 2. Here didacticians could be seen 
clearly to take on board teachers' perspectives and to build these into ongoing 
activity in workshops. Increasingly in Phase 2, input from teachers relating to 
activity in classrooms became a central feature of plenary sessions. Curriculum 
topics were used explicitly as a focus for mathematical activity. Same- level groups 
predominated. Feedback from a focus group interview with each school team at the 
end of Phase 2 indicated that teachers had appreciated didacticians' 
accommodation to their perspectives in a range of factors and showed 
corresponding activity in classrooms. Invitations from teachers to didacticians to 
videorecord innovative activity in classrooms resulted in a bank of videodata 
charting development in classrooms. We might see, in retrospect the meeting 
between Phases 1 and 2 as a watershed in project activity. EngestrSm's (1999) 
theory of expansive learning might be seen to capture this watershed. 

Expansive Learning 

The outcome of tensions, such as those expressed above, in the LCM project was 
that activity went on. We did not see a breakdown. Trust, good will and positive 
intentions led to realization (both recognition and making-real) of ways of working 
that enabled some achievement of some goals (on both sides) to some extent. In 
this process, there was some event or initiative which acted as a force to resolve 
tensions - expressed by EngestrSm (1999) as "expansive learning". For example, 
during the first phase we had seen a build up of tension as teachers engaged with 
activity, provided clear evidence of valuing the project and their participation, yet 

9 The LCM project was funded for four years. During this time, there were three Phases of activity, each 
of one school year, in which didacticians and teachers worked together as described here. 



at the same time increasingly expressed a wish for modified forms of action (such 
as the nature of small groups or the kinds of mathematical tasks). The meeting 
between the phases allowed overt expression of desire for alternative action and 
clear suggestions for the form such action might take. 

Expansive learning is rooted in the activity theory concepts expressed above - 
notably goal directed mediated action, based in Vygotsky (1978) and Leont'ev 
(1979). EngestrSm (1999, p. 382), following Leont'ev (1979), expresses it as a 
dialectic of "ascending from the abstract to the concrete" and adds (pp. 382-383): 

A method of grasping the essence of an object by tracing and reproducing 
theoretically the logic of its development, of its historical formation through 
the emergence and resolution of inner contradictions. [...] The initial simple 
idea is transformed into a complex object, a new form of practice. [...] The 
expansive cycle begins with individual subjects questioning the accepted 
practice, and it gradually expands into a collective movement or institution. 

Through complex interactions traceable to all three elements of the hidden 
curriculum, participants in the project are able to recognize and isolate the inner 
contradictions expressed by Engestrdm (1999). In the case above, the concerns 
about groups and about tasks in workshops, through the between-phases meeting, 
led to the emergence of the new idea of planlegge et opplegg through which 
planning in homogenous groups in a workshop with didacticians' support could 
lead to teachers having suitable activity for their classrooms leading to 
development for students' learning of mathematics. What started as internal 
rumblings within activity resulted in an external development explicit for all to 
engage with. Such analysis enables us to trace our activity, noting its historical 
development and becoming clearer about the issues, tensions or contradictions 
inherent in the developmental process. 

As a further example, I refer to an event with took place in Phase 3 of the 
project. This phase was introduced (by didacticians) as focusing on declaring and 
achieving school goals for development of mathematics learning and teaching 
within a school. Activity in Phase 3 proceeded along familiar lines with 
engagement in workshops and associated work in schools and with associated 
issues and tensions acknowledged but not resolved. The focus on school goals was 
elusive and progression towards school goals not achieved. Then one didactician 
suggested a task that was to have important consequences for the goals of Phase 3. 
The task was connected to a series of three workshops focusing on algebra. It 
involved teachers in undertaking some focused observation of some of their own 
students related to work on algebra. Teachers were asked to bring to the next 
workshop some input from their observations. The workshop was organized to 
develop a "red thread" through observations at different levels of students' 
algebraic understanding across the range of school levels. In order to visualize 
teachers' judgement on the quality of students' algebraic thinking, teachers were 
asked to pin their written observations to a line which was strung across the 
workshop's main room. The coffee break allowed all to view the line and think 
about its contents. The quality of teachers' perceptions expressed in the final 



plenary discussion and comments received from several teachers after the 
workshop indicated that this had been an important experience for teachers: most 
significant had been their insights into the thinking and understanding of their own 
students and recognition of the task as a research event with serious learning 
outcomes for themselves. The task had provided the opportunity for expansion and 
for a breakthrough in activity. 

Teachers' participation and comments in and from this activity suggested to 
didacticians that certain goals had been achieved. Teachers had engaged overtly in 
a research task, conducting activity in their schools, findings the time to do so, 
recognizing their learning, and valuing their insights into students' 
perceptions/thinking/understandings of algebra. This signified for didacticians 
strong developmental outcomes from their own activity and participation - with 
evidence of both teachers' learning and didacticians' associated learning. For 
example, teachers suddenly came to see, through their study of students' thinking 
and activity in algebra, how they could explore in their school environment ways to 
develop teaching and learning; didacticians saw the nature of a task that could lead 
to teachers' effective recognition of the nature of school goals for students' 
development and learning in mathematics. 

Seeing the enterprise in terms of an activity system made it possible to pick out 
elements in the complexity and trace developmental patterns for participants in the 
project (see Goodchild & Jaworski, 2005; Jaworski & Goodchild, 2006). In this 
process, tensions became evident as catalysts providing opportunity for learning. 
We see the nature of community as central to this provision of opportunity. During 
the three years, the members of the project community came to know each other as 
colleagues, appreciating good intentions, trusting good will, recognizing 
differences, respecting alternative points of view and becoming aware of 
developing thinking and associated possibility for action. This is not to claim 
hugely visible changes to the everyday practices in which established communities 
were rooted, but rather to recognize a relationship between developmental aims 
and the realities of normal working life. For example, the structure in a school 
could not change to suit the aims of the project; nor should the project fail because 
these aims could not be met. So, in what ways might we accommodate to achieve 
the aims? Through such recognition also, the aims become more understandable 
and perhaps more open to flexibility in their achievement - that is we were able to 
relate the aims to real settings and work out alternative approaches compatible with 
the aims. 


This chapter has interwoven complex aspects of theory and a specific 
developmental research project to illuminate notions of the development of 



mathematics learning and teaching through developmental research in inquiry 
communities involving teachers and didacticians. 10 

In this final section of the chapter my purpose is to pull out to a more general 
viewpoint on communities of inquiry and the associated theoretical perspectives. 
Key areas of theory have been 

• Communities of practice with notions of belonging through 
engagement, imagination and alignment (Wenger, 1998) shifting to 
critical alignment through inquiry (Jaworski, 2006); 

• Mediated activity between people involving individuals acting with 
mediational means (Vygotsky, 1978; Wertsch, 1991); 

• The motivated nature of activity involving goal-directed action 
(Leont'ev, 1979); 

• Engestrom's expanded mediational triangle and the concept of 
expansive learning (EngestrBm, 1998, 1999). 

The concept of community is clearly central in all of these and needs no further 
comment. The place of inquiry perhaps needs further elucidation. Inquiry brings 
the critical element to community of practice through which participants can 
inquire into existing practices with possibility to modify and improve. Inquiry can 
be seen as a mediational tool in social settings enabling development of knowing 
between people and hence of participative individuals. Inquiry as in the 
design/inquiry cycle promotes goal-directed action leading to developmental 
outcomes. Inquiry ways of being allow the possibility of contradictions emerging 
as powerful motivators for expansion within an activity system. 

The inquiry community starts with intentions to use inquiry as a tool for learning 
and development. Through engagement with an inquiry cycle in the design of tasks 
and opportunity for participation, a community grows into inquiry ways of being 
which encourage mediation of complexity within the hidden curriculum of systems 
and structures that constrain development. As compared to established 
communities of practice, in which norms of practice nurture undesirable states, the 
inquiry community is emergent. It does not avoid issues, tensions and 
contradictions, but deals with them as part of emergent recognition and 
understanding leading to possibilities for expansive learning. Inquiry ways of being 
accept the unfinished nature of learning and development. There is not an end 

10 For those interested in knowing more about the LCM project, the website 
contains a list of relevant publications and the book (Jaworski, Fuglestad, Bjuland, Breiteig, Goodchild, 
& Grevholm, 2007) charts the project as a whole. 




LCM ended in December 2007. However, in 2006, an extension to LCM was 
already started in the form of a n